IE ivigjc iianics i^e Jt. THEORETICAL ELEMENTS OF ELECTRICAL ENGINEERING < \MQ QraW'3/illBook & 7m PUBLISHERS OF BOOKS F O R^ Coal Age v Electric Railway Journal Electrical World v Engineering News-Record American Machinist >v The Contractor Engineering 8 Mining Journal <* Power Metallurgical 6 Chemical Engineering Electrical Merchandising THEORETICAL ELEMENTS OF ELECTRICAL ENGINEERING BY CHARLES PROTEUS STEINMETZ, A.M., PH.D. FOURTH EDITION THOROUGHLY REVISED AND ENTIRELY RESET FOURTH IMPRESSION McGRAW-HILL BOOK COMPANY, INC 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., LTD. 6 & 8 BOUVERIE ST., E. C. 1915 e- Engineering library COPYRIGHT, 1909, 1915, BY THE MCGRAW-HILL BOOK COMPANY, INC. THE MAPI.E PRTJSS YORK PA PREFACE TO FIRST EDITION THE first part of the following volume originated from a series of University lectures which I once promised to deliver. This part can, to a certain extent, be considered as an intro- duction to my work on "Theory and Calculation of Alternating Current Phenomena," leading up very gradually from the ordi- nary sine wave representation of the alternating current to the graphical representation by polar coordinates, from there to rectangular components of polar vectors, and ultimately to the symbolic representation by the complex quantity. The present work is, however, broader in its scope, in so far as it comprises the fundamental principles not only of alternating, but also of direct currents. The second part is a series of monographs of the more impor- tant electrical apparatus, alternating as well as direct current. It is, in a certain respect, supplementary to "Alternating Current Phenomena." While in the latter work I have presented the general principles of alternating current phenomena, in the pres- ent volume I intended to give a specific discussion of the par- ticular features of individual apparatus. In consequence thereof, this part of the book is somewhat less theoretical, and more descriptive, my intention being to present the most important electrical apparatus in all their characteristic features as regard to external and internal structure, action under normal and ab- normal conditions, individually and in connection with other ap- paratus, etc. I have restricted the work to those apparatus which experi- ence has shown as of practical importance, and give only those theories and methods which an extended experience in the de- sign and operation has shown as of practical utility. I con- sider this the more desirable as, especially of late years, electri- cal literature has been haunted by so many theories' (for instance of the induction machine) which are incorrect, or too compli- cated for use, or valueless in practical application. In the class last mentioned are most of the graphical methods, which, while they may give an approximate insight in the inter-relation of 520444 vi ' PREFACE phenomena, fail entirely in engineering practice owing to the great difference in the magnitudes of the vectors in the same diagram, and to the synthetic method of graphical representa- tion, which generally require one to start with the quantity which the diagram is intended to determine. I originally intended to add a chapter on Rectifying Apparatus, as arc light machines and alternating current rectifiers, but had to postpone this, due to the incomplete state of the theory of these apparatus. The same notation has been used as in the Third Edition of " Alternating Current Phenomena," that is, vector quantities denoted by dotted capitals. The same classification and nomen- clature have been used as given by the report of the Standardiz- ing Committee of the American Institute of Electrical Engineers. CHARLES PROTEUS STEINMETZ. SCHENECTADY, N. Y., May 1st, 1901. PREFACE TO THIRD EDITION NEARLY eight years have elapsed since the appearance of the second edition, during which time the book has been reprinted without change, and a revision, therefore, became greatly desired. It was gratifying, however, to find that none of the contents of the former edition had to be dropped as superseded or anti- quated. However, very much new material had to be added. During these eight years the electrical industry has progressed at least as rapidly as in any previous period, and apparatus and phenomena which at the time of the second edition were of theoretical interest only, or of no interest at all, have now as- sumed great industrial importance, as for instance the single- phase commutator motor, the control of commutation by corn- mutating poles, etc. Besides rewriting and enlarging numerous paragraphs through- out the text, a number of new sections and chapters have been added, on alternating-current railway motors, on the control of commutation by commutating poles ("interpoles"), on con- verter heating and output, on converters with variable ratio of conversion (" split-pole converters"), on three- wire generators and converters, short-circuit currents of alternators, stability and regulation of induction motors, induction generators, etc. In conformity with the arrangement used in my other books, the paragraphs of the text have been numbered for easier refer- ence and convenience. When reading the book, or using it as text-book, it is recom- mended : After reading the first or general part of the present volume, to read through the first 17 chapters of "Theory and Calculation of Alternating Current Phenomena/' omitting, however, the mathematical investigations as far as not absolutely required for the understanding of the text, and then to take up the study of the second part of the present volume, which deals with special apparatus. When reading this second part, it is recom- mended to parallel its study with the reading of the chapter of " Alternating Current Phenomena" which deals with the same vii viii PREFACE subject in a different manner. In this way a clear insight into the nature and behavior of apparatus will be imparted. Where time is limited, a large part of the mathematical dis- cussion may be skipped and in that way a general review of the material gained. Great thanks are due to the technical staff of the McGraw- Hill Book Company, which has spared no effort to produce the third edition in as perfect and systematic a manner as possible, and to the numerous engineers who have greatly assisted me by pointing out typographical and other errors in the previous edition. CHARLES PROTEUS STEINMETZ. SCHENECTADY, September, 1909. PREFACE TO THE FOURTH EDITION With the fourth edition, " Theoretical Elements of Electrical Engineering" has been radically revised and practically rewritten. Since 1897 and 1898, when the first editions of " Alternating Current Phenomena" and "Theoretical Elements" appeared, electrical engineering has enormously expanded and diversified. New material thus had to be added to the successive editions until now it has become utterly impossible to deal with the sub- ject matter adequately within the limited scope of the two books. Therefore in the present edition everything beyond the most fundamental principles of general theory and special ap- paratus has been withdrawn, to make room for the adequate rep- resentation of the theoretical elements of present-day electrical engineering. The same will be done in the new edition of "Alter- nating Current Phenomena," which is in preparation, and the material, which thus does not find room any more in these two books, together with such additional matters as the development of electrical engineering requires, will be collected in a third volume. In the present edition, the crank diagram of vector represen- tation, and the symbolic method based on it, which denotes the inductive impedance by Z r + jx, has been adopted in con- formity with the decision of the International Electrical Congress of Turin. This crank diagram is somewhat inferior in utility to the polar diagram used in the previous editions, since it is limited to sine waves. I believe it was adopted without sufficient consideration of the relative merits. Nevertheless the advan- tage of the use of the same vector representation in all elementary text-books on electrical engineering, seems to me to outweigh the advantage of the polar diagram resulting from its ability to rep- resent waves which are not sines, while in advanced electrical engineering both representations will have to remain in use. CHARLES P. STEINMETZ. SCHENECTADY, N. Y., October, 1915. IX CONTENTS PART I GENERAL THEORY PAGE 1. Magnetism and Electric Current. 1 2. Magnetism and E.M.F. 9 3. Generation of E.M.F. 11 4. Power and Effective Values. 15 5. Self-Inductance and Mutual Inductance. 21 6. Self-Inductance of Continuous-Current Circuits. 24 7. Inductance in Alternating-Current Circuits. 31 8. Power in Alternating-Current Circuits. 39 9. Vector Diagrams. 41 10. Hysteresis and Effective Resistance. 48 11. Capacity and Condensers. 54 12. Impedance of Transmission Lines. 57 13. Alternating-Current Transformer. 67 14. Rectangular Coordinates. 77 15. Load Characteristic of Transmission Line. 85 16. Phase Control of Transmission Lines. 90 17. Impedance and Admittance. 98 18. Equivalent Sine Waves. 106 19. Fields of Force. 20. Nomenclature. 118 PART II SPECIAL APPARATUS INTRODUCTION. 121 A. SYNCHRONOUS MACHINES. I. General. 126 II. Electromotive Forces. 128 III. Armature Reaction. 130 IV. Self-Inductance. 133 V. Synchronous Reactance. 136 VI. Characteristic Curves of Alternating-Current Generator. 138 VII. Synchronous Motor. 141 VIII. Characteristic Curves of Synchronous Motor. 143 IX. Magnetic Characteristic or Saturation Curve. 147 X. Efficiency and Losses. 149 XI. Unbalancing of Polyphase Synchronous Machines. 150 XII. Starting of Synchronous Motors. 151 Xin. Parallel Operation. 153 XIV. Division of Load in Parallel Operation. 154 xi xii CONTENTS SYNCHRONOUS MACHINES (continued). PAGE XV. Fluctuating Cross-Currents in Parallel Operation. 155 XVI. High Frequency; Cross-Currents between Synchronous Machines. 159 XVII. Short-Circjiit Currents of Alternators. 160 B. DIRECT-CURRENT COMMUTATING MACHINES. I. General. 166 II. Armature Winding. 168 III. Generated Electromotive Forces. 177 IV. Distribution of Magnetic Flux. 178 V. Effect of Saturation on Magnetic Distribution. 182 VI. Effect of Commutating Poles. 184 VII. Effect of Slots on Magnetic Flux. 190 VIII. Armature Reaction. 193 IX. Saturation Curves. 194 X. Compounding. 196 XI. Characteristic Curves. 198 XII. Efficiency and Losses. 198 XIII. Commutation. 199 XIV. Types of Commutating Machines. 206 A. Generators. Separately excited and Magneto, Shunt, Series, Compound. 208 B. Motors. Shunt, Series, Compound. 215 XV. Appendix. Alternating-Current Commutator Motor. 218 C. SYNCHRONOUS CONVERTERS. I. General. 223 II. Ratio of E.M.Fs. and of Currents. 224 III. Variation of the Ratio of E.M.Fs. 231 IV. Armature Current and Heating. 232 V. Armature Reaction. 245 VI. Reactive Currents and Compounding. 250 VII. Variable Ratio Converters (Split-Pole Converters). 252 V11I. Starting. . 253 IX. Inverted Converters. 255 X. Frequency. 257 XI. Double-Current Generators. 259 XII. Conclusion. 261 XIII. Appendix. Direct-Current Converter. 262 XIV. Three- Wire Generator and Converter. 270 D. ALTERNATING-CURRENT TRANSFORMER. I. General. 277 II. Excitation. 279 CONTENTS xiii ALTERNATING-CURRENT TRANSFORMER (continued). PAGE III. Losses and Efficiency. 280 IV. Regulation. 285 V. Short-Circuit Current. 293 VI. Heating and Ventilation. 294 VII. Types of Transformers. 295 VIII. Auto-Transformers. 299 IX. Reactors. 302 E. INDUCTION MACHINES. I. General. 306 II. Polyphase Induction Motor. 1. Introduction. 310 2. Calculation. 311 3. Load and Speed Curves. 317 4. Effect of Armature Resistance and Starting. 322 III. Single-phase Induction Motor. 1. Introduction. 326 2. Load and Speed Curves. 329 3. Starting Devices of Single-phase Motors. 333 4. Acceleration with Starting Device. 338 IV. Induction Generator. 1. Introduction. 340 2. Constant Speed Induction or Asynchronous Generator. 342 3. Power Factor of Induction Generator. 343 V. Induction Booster. 349 VI. Phase Converter. 351 VII. Frequency Converter or General Alternating-Current Trans- former. 354 VIII. Concatenation of Induction Motors. 359 INDEX 363 PART I GENERAL THEORY 1. MAGNETISM AND ELECTRIC CURRENT 1. A magnet pole attracting (or repelling) another magnet pole of equal strength at unit distance with unit force 1 is called a unit magnet pole. The space surrounding a magnet pole is called a magnetic field of force, or magnetic field. The magnetic field at unit distance from a unit magnet pole is called a unit magnetic field, and is represented by one line of magnetic force (or shortly "one line") per square centimeter, and from a unit magnet pole thus issue a total of 4 TT lines of magnetic force. The total number of lines of force issuing from a magnet pole is called its magnetic flux. The magnetic flux $ of a magnet pole of strength m is, < = 4 irm. At the distance I from a magnet pole of strength m, and therefore of flux $> = 4 xw, assuming a uniform distribution in all directions, the magnetic field has the intensity, n - since the 3> lines issuing from the pole distribute over the area of a sphere of radius I, that is, the area 4 irl 2 . A magnetic field of intensity H exerts upon a magnet pole of strength m the force, mH. Thus two magnet poles of strengths mi and m z , and distance I from each other, exert upon each other the force, 1 That is, at 1 cm. distance with such force as to give to the mass of 1 gram the acceleration of 1 cm. per second. 1 2 *:LE' VENTS OF ELECTRICAL ENGINEERING 2. Electric currents produce magnetic fields also; that is, the space surrounding the conductor carrying an electric current is a magnetic field, which appears and disappears and varies with the current producing it, and is indeed an essential part of the phenomenon called an electric current. Thus an electric current represents a' magnetomotive force (m.m.f.). The magnetic field of a straight conductor, whose return conductor is so far distant as not to affect the field, consists of lines of force surrounding the conductor in concentric circles. The intensity of this magnetic field is directly proportional to the current strength and inversely proportional to the dis- tance from the conductor. Since the lines of force of the magnetic field produced by an electric current return into themselves, the magnetic field is a magnetic circuit. Since an electric current, at least a steady current, can exist only in a closed circuit, electricity flows in an electric circuit. The magnetic circuit produced by an electric current surrounds the electric circuit through which the electricity flows, and inversely. That is, the electric circuit and the magnetic circuit are interlinked with each other. Unit current in an electric circuit is the current which produces in a magnetic circuit of unit length the field intensity 4?r, that is, produces as many lines of force per square centimeter as issue from a unit magnet pole. In unit distance from an electric conductor carrying unit current, that is, in a magnetic circuit of length 2?r, the field 4-7T intensity is ~ 2, and in the distance 2 the field intensity is unity; that is, unit current is the current which, in a straight conductor, whose return conductor is so far distant as not to affect its magnetic field, produces field intensity 2 in unit distance from the conductor. One-tenth of unit current is the practical unit, called one ampere. 3. One ampere in an electric circuit or turn, that is, one ampere-turn, thus produces in a magnetic circuit of unit length the field intensity 0.4 w, and in a magnetic circuit of length 0.4 TT I the field intensity '-j , and F ampere-turns produce in a magnetic circuit of length I the field intensity: 0.4 irF v tf H = j lines of force per sq. cm. MAGNETISM AND ELECTRIC CURRENT 3 regardless whether the F ampere-turns are due to F amperes F in a single turn, or 1 amp. in F turns, or amperes in n turns. F, that is, the product of amperes and turns, is called magneto- motive force (m.m.f.). The m.m.f. per unit length of magnetic circuit, or ratio, _ _ m.m.f. _ ' " " length of magnetic circuit is called the magnetizing force, or magnetic gradient. Hence, m.m.f. is expressed in ampere-turns; magnetizing force in ampere-turns per centimeter (or in practice frequently ampere-turns per inch), field intensity in lines of magnetic force per square centimeter. At the distance I from the conductor of a loop or circuit of F ampere-turns, whose return conductor is so far distant as not to affect the field, assuming the m.m.f. = F, since the length of the magnetic circuit = 2 irl, we obtain as the magnetizing force, : '- and as the field intensity, 2 F H = 0.4 TT = 2^- 4. The magnetic field of an electric circuit consisting of two parallel conductors (or any number of conductors, in a poly- phase system), as the two wires of a transmission line, can be considered as the superposition of the separate fields of the conductors (consisting of concentric circles). Thus, if there are I amperes in a circuit consisting of two parallel conductors (conductor and return conductor), at the distance li from the first and h from the second conductor, the respective field intensities are, - l ~ ~TT and "-T and the resultant field intensity, if r = angle between the direc- tions of the two fields, H = \/# i 2 + # 2 2 + 2 #!# 2 COST, COST. 4 ELEMENTS OF ELECTRICAL ENGINEERING In the plane of the conductors, where the two fields are in the same or opposite direction, the resultant field intensity is, 7 7 M2 where the plus sign applies to the space between, the minus sign the space outside of the conductors. The resultant field of a circuit of parallel conductors con- sists of excentric circles, interlinked with the conductors, and crowded together in the space between the conductors as shown in Fig. 1 by drawn lines. FIG. 1. Magnetic field of parallel conductors. The magnetic field in the interior of a spiral (solenoid, helix, coil) carrying an electric current consists of straight lines. 5. If a conductor is coiled in a spiral of I centimeter axial N length of spiral, and N turns, thus n = j- turns per centimeter length of spiral, and / = current, in amperes, in the conductor, the m.m.f. of the spiral is F = IN, and the magnetizing force in the middle of the spiral, assuming, the latter of very great length, T N T f = nI = T I, thus the field intensity in the middle of the spiral or solenoid, H = 0.4 TT/ = 0.4 MAGNETISM AND ELECTRIC CURRENT 5 Strictly this is true only in the middle part of a spiral of such length that the m.m.f. consumed by the external or mag- netic return circuit of the spiral is negligible compared with the m.m.f. consumed by the magnetic circuit in the interior of the spiral, or in an endless spiral, that is, a spiral whose axis curves back into itself, as a spiral whose axis is curved in a circle. Magnetomotive force F applies to the total magnetic circuit, or part of the magnetic circuit. It is measured in ampere- turns. Magnetizing force / is the m.m.f. per unit length of mag- netic circuit. It is measured in ampere-turns per centimeter. Field intensity H is the number of lines of force per square centimeter. If I = length of the magnetic circuit or a part of the magnetic circuit, F = V, f = j, H = 0.4 / / = H 0.47T' = 1.257/ /= 0.796 #. 6. The preceding applies only to magnetic fields in air or other unmagnetic materials. If the medium in which the magnetic field is established is a "magnetic material," the number of lines of force per square centimeter is different and usually many times greater. (Slightly less in diamagnetic materials.) The ratio of the number of lines of force in a medium, to the number of lines of force which the same magnetizing force would produce in air (or rather in a vacuum), is called the permeability or magnetic conductivity /* of the medium. The number of lines of force per square centimeter in a mag- netic medium is called the magnetic induction B. The number of lines of force produced by the same magnetizing force in air, or rather, in the vacuum, is called the field intensity H. In air, magnetic induction B and field intensity H are equal. As a r-ule, the magnetizing force in a magnetic circuit is changed by the introduction of the magnetic material, due to the change of distribution of the magnetic flux. The permeability of air = 1 and is constant. 6 ELEMENTS OF ELECTRICAL ENGINEERING . The permeability of iron and other magnetic materials varies with the magnetizing force between a little above 1 and values beyond 10,000 in soft iron. The magnetizing force / in a medium of permeability /* pro- duces the field intensity H = 0.4 irf and the magnetic induction B = 0.4 TTflf. EXAMPLES 7. (1) A pull of 2 grams at 4 cm. radius is required to hold a horizontal bar magnet 12 cm. in length, pivoted at its center, in a position at right angles to the magnetic meridian. What is the intensity of the poles of the magnet, and the number of lines of magnetic force issuing from each pole, if the horizontal intensity of the terrestrial magnetic field H = 0.2, and the acceleration of gravity = 980? The distance between the poles of the bar magnet may be assumed as five-sixths of its length. Let m = intensity of magnet poles. I = 5 is the radius on which the terrestrial magnetism acts. Thus 2mHl = 2'm = torque exerted by the terrestrial magnetism. 2 grams weight = 2 X 980 = 1960 units of force. These at 4 cm. radius give the torque 4 X I960 = 7840 g cm. Hence 2m = 7840. m = 3920 is the strength of each magnet pole and 3> = 4 Trm = 49,000, the number of lines of force issuing from each pole. 8. (2) A conductor carrying 100 amp. runs in the direc- tion of the magnetic meridian. What position will a compass needle assume, when held below the conductor at a distance of 50 cm., if the intensity of the terrestrial magnetic field is 0.2? The intensity of the magnetic field of 100 amp., 50 cm. 027 100 from the conductor, is H = ^ = 0.2 X -^r = 0.4, the direc- tion is at right angles to the conductor, that is, at right angles to the terrestrial magnetic field. If T = angle between compass needle and the north pole of the magnetic meridian, 1 = length of needle, m = intensity of its magnet pole, the torque of the terrestrial magnetism is Hmlo sin T = 0.2 mlo sin r, the torque of the current is 0.2/raZocosr cos r = - - = 0.4 MAGNETISM AND ELECTRIC CURRENT 7 In equilibrium, 0.2 ml Q sin r = 0.4 ml Q cos r, or tan r = 2, r = 63.4. 9. (3) What is the- total magnetic flux per I = 1000 m. length, passing between the conductors of a long distance transmission line carrying 7 amperes of current, if Id = 0.82 cm. is the diam- eter of the conductors (No. B. & S.), 1 8 = 45 cm. the spacing or distance between them? FIG. 2. Diagram of transmission line for inductance calculation. At distance l r from the center of one of the conductors (Fig. 2), the length of the magnetic circuit surrounding this conductor is 2irl r) the m.m.f., 7 ampere-turns; thus the magnetizing force / = s r> and the field intensity H = 0.4 irf = -^ > and the A TTl r L r flux in the zone dl r is d$ = - j -> and the total flux from the surface of the conductor to the next conductor is, ).2 Ildl r 0.27/Flog e Z r j = 0.2/nog.^- The same flux is produced by the return conductor in the same direction, thus the total flux passing between the trans- mission wires is, 2 $ = 0.4 II log, p I'd or per 1000 m. = 10 5 cm. length, QO 2 $ = 0.4 X 10 5 / log, TT^ = 0.4 X 10 5 X 4.70 I = 0.188 X 10 6 7, 8 ELEMENTS OF ELECTRICAL ENGINEERING or 0.188 / megalines or millions of lines per line of 1000 m. of which 0.094 / megalines surround each of the two conductors. 10. (4) In an alternator each pole has to carry 6.4 millions of lines, or 6.4 megalines magnetic flux. How many ampere- turns per pole are required to produce this flux, if the magnetic 2 8 10 2 14 6 FIG. 3. Magnetization curves of various irons. circuit in the armature of laminated iron has the cross section of 930 sq. cm. and the length of 15 cm., the air-gap between stationary field poles and revolving armature is 0.95 cm. in length and 1200 sq. cm. in section, the field pole is 26.3 cm. in length and 1075 sq. cm. in section, and is of laminated iron, MAGNETISM AND E.M.F. 9 and the outside return circuit or yoke has a length per pole of 20 cm. and 2250 sq. cm. section, and is of cast iron? The magnetic densities are: BI = 6880 in the armature, B 2 = 5340 in the air-gap, B 3 = 5950 in the field pole, and B 4 = 2850 in the yoke. The permeability of sheet iron is m = 2550 at 5i = 6880, MS = 2380 at B 3 = 5950. The permeability of cast iron is /z 4 = 280 at B 4 = 2850. Thus the field intensity/ H = - j is: Hi = 2.7, # 2 = 5340, H 3 = 2.5, H, = 10.2. The magnetizing force (/ = {TT ) is > f l = 2 ' 15 > ^ 2 = 4250 ' / 3 = 1.99, / 4 = 8.13 ampere-turns per centimeter. Thus the m.m.f. (F = fl) is: Fi = 32, F 2 = 4040, F 3 = 52, F 4 = 163, or the total m.m.f. per pole is F = Fi + F 2 + F 3 + ^4 = 4290 ampere-turns. The permeability p of magnetic materials varies with the density B, thus tables or curves have to be used for these quan- tities. Such curves are usually made out for density B and magnetizing force /, so that the magnetizing force / correspond- ing to the density B can be derived directly from the curve. Such a set of curves is given in Fig. 3. 2. MAGNETISM AND E.M.F. 11. In an electric conductor moving relatively to a magnetic field, an e.m.f. is generated proportional to the rate of cutting of the lines of magnetic force by the conductor. Unit e.m.f. is the e.m.f. generated in a conductor cutting one line of magnetic force per second. 10 8 times unit e.m.f. is the practical unit, called the volt. Coiling the conductor n fold increases the e.m.f. n fold, by cutting each line of magnetic force n times. In a closed electric circuit the e.m.f. produces an electric current. The ratio of e.m.f. to electric current produced thereby is called the resistance of the electric circuit. Unit resistance is the resistance of a circuit in which unit e.m.f. produces unit current. 10 9 times unit resistance is the practical unit, called the ohm. 10 ELEMENTS OF ELECTRICAL ENGINEERING The ohm is the resistance of a circuit, in which 1 volt produces 1 amp. The resistance per unit length and unit section of a conductor is called its resistivity, p. The resistivity p is a constant of the material, varying with the temperature. The resistance r of a conductor of length I, area or section A, ... lp and resistivity p is r = -7" 12. If the current in the electric circuit changes, starts, or stops, the corresponding change of the magnetic field of the current generates an e.m.f in the conductor carrying the current, which is called the e.m.f. of self-induction. If the e.m.f. in an electric circuit moving relatively to a magnetic field produces a current in the circuit, the magnetic field produced by this current is called its magnetic reaction. The fundamental law of self-induction and magnetic reaction is that these effects take place in such a direction as to oppose their cause (Lentz's law). Thus the e.m.f. of self-induction during an increase of current is in the opposite direction, during a decrease of current in the same direction as the e.m.f. producing the current. The magnetic reaction of the current produced in a circuit moving out of a magnetic field is in the same direction, in a circuit moving into a magnetic field in opposite direction to the magnetic field. Essentially, this law is nothing but a conclusion from the law of conservation of energy. EXAMPLES 13. (1) An electromagnet is placed so that one pole sur- rounds the other pole cylindrically as shown in section in Fig. 4, and a copper cylinder revolves between these poles at 3000 rev. per min. What is the e.m.f. generated between the ends of this cylinder, if the magnetic flux of the electromagnet is < = 25 megalines? During each revolution the copper cylinder cuts 25 mega- lines. It makes 50 rev. per sec. Thus it cuts 50 X 25 X 10 6 = 12.5 X 10 8 lines of magnetic flux per second. Hence the gener- ated e.m.f. is E = 12.5 volts. GENERATION OF E.M.F. 11 Such a machine is called a " unipolar," or more properly a " non-polar" or an "acyclic," generator. 14. (2) The field spools of the 20-pole alternator in Section 1, Example 4, are wound each with 616 turns of wire No. 7 (B. & S.), 0.106 sq. cm. in cross section and 160 cm. mean length of turn. The 20 spools are connected in series. How many amperes and how many volts are required for the excitation of this alternator field, if the resistivity of copper is 1.8 X 10~ 6 ohms per cm. 3 1 FIG. 4. Unipolar generator. Since 616 turns on each field spool are used, and 4280 ampere- 4280 turns required, the current is fi1fi = 6.95 amp. The resistance of 20 spools of 616 turns of 160 cm. length, 0.106 sq. cm. section, and 1.8 X 10~ 6 resistivity is, 20 X 616 X 160 X 1.8 X 10~ 6 = 33.2 ohms, 0.106 and the e.m.f. required, 6.95 X 33.2 = 230 volts. i 3. GENERATION OF E.M.F. 15. A closed conductor, convolution or turn, revolving in a magnetic field, passes during each revolution through two positions of maximum inclosure of lines of magnetic force A in Fig. 5, and two positions of zero inclosure of lines of mag- netic force B in Fig. 5. 1 cm. 3 refers to a cube whose side is 1 cm., and should not be confused with cu. cm. 12 ELEMENTS OF ELECTRICAL ENGINEERING Thus it cuts during each revolution four times the lines of force inclosed in the position of maximum inclosure. If 3> = the maximum number of lines of force inclosed by the conductor, / = the frequency in revolutions per second or cycles, and n = number of convolutions or turns of the con- ductor, the lines of force cut per second by the conductor, and thus the average generated e.m.f. is, E = 4 fn$ absolute units, = 4fn3> ID" 8 volts. FIG. 5. Generation of e.m.f. If / is given in hundreds of cycles, < in megalines, E = 4n$ volts. If a coil revolves with uniform velocity through a uniform magnetic field, the magnetism inclosed by the coil at any instant is, $ COS T where $ = the maximum magnetism inclosed by the coil arid T = angle between coil and its position of maximum inclosure of magnetism. The e.m.f. generated in the coil, which varies with the rate of cutting or change of $ cos T, is thus, e = EQ sin T, where E Q is the maximum value of e.m.f., which takes place for T = 90, or at the position of zero inclosure of magnetic flux since in this position the rate of cutting is greatest. 2 Since avg. (sin T) = -, the average generated e.m.f. is, GENERATION OF E.M.F. 13 Since, however, we found above that E = 4 fn& is the average generated e.m.f., it follows that EQ = 2 irfn& is the maximurft, and e = 2 7r/n$ sin r the instantaneous generated e.m.f. The interval between like poles forms 360 electrical-space de- grees, and in the two-pole model these are identical with the mechanical-space degrees. With uniform rotation, Fig. 6, the space angle, r, is proportional to time. Time angles are designated by 6, and with uniform rotation & = r, T being measured in elec- trical-space degrees. FIG. 6. Generation of e.m.f. by rotation. The period of a complete cycle is 360 time degrees, or 2 TT or -- seconds. In the two-pole model the period of a cycle is that of one complete revolution, and in a 2 n p -pole machine, of that n p of one revolution. Thus, 6 = 2 wft e = 2>jrfn3> sin 2 irft. If the time is not counted from the moment of maximum Lnclosure of magnetic flux, but ti = the time at this moment, we have e = 2 irfn$ sin 2 IT} (t - ti) or, e = 2-jrfn& sin (6 0i), where 0i = 2 irfli is the angle at which the position of maxi- mum inclosure of magnetic flux takes place, and is called its phase. These e.m.fs. are alternating. If at the moment of reversal of the e.m.f. the connections between the coil and the external circuit are reversed, the e.m.f. in the external circuit is pulsating between zero and E Q , but has the same average value E. If a number of coils connected in series follow each other H ELEMENTS OF ELECTRICAL ENGINEERING successively in their rotation through the magnetic field, as the armature coils of a direct-current machine, and the connections of each coil with the external circuit are reversed at the moment of reversal of its e.m.f., their pulsating e.m.fs. superimposed in the external circuit make a more or less steady or continuous external e.m.f. The average value of this e.m.f. is the sum of the average values of the e.m.fs. of the individual coils. Thus in a direct-current machine, if $ = maximum flux in- closed per turn, n = total number of turns in series from com- mutator brush to brush, and / = frequency of rotation through the magnetic field. E = 4/n$> = generated e.m.f. ($ in megalines, / in hundreds of cycles per second). This is the formula of the direct-current generator. EXAMPLES 17. (1) A circular wire coil of 200 turns and 40 cm. mean diameter is revolved around a vertical axis. What is the horizontal intensity of the magnetic field of the earth, if at a speed of 900 rev. per min. the average e.m.f generated in the coil is 0.028 volt? 40 2 7T The mean area of the coil is j = 1255 sq. cm., thus the terrestrial flux inclosed is 1255 H, and at 900 rev. per min. or 15 rev. per sec., this flux is cut 4 X 15 = 60 times per second by each turn, or 200 X 60 = 12,000 times by the coil. Thus the total number of lines of magnetic force cut by the conductor per second is 12,000 X 1255 H = 0.151 X 10 8 H, and the average generated e.m.f. is 0.151 H volts. Since this is = 0.028 volt, H = 0.186. 18. (2) In a 550-volt direct-current machine of 8 poles and drum armature, running at 500 rev. per min., the average vol- tage per commutator segment shall not exceed 11, each armature coil shall contain one turn only, and the number of commutator segments per pole shall be divisible by 3, so as to use the machine as three-phase converter. What is the magnetic flux per field pole? 550 volts at 11 volts per commutator segment gives 50, or as next integer divisible by 3, n = 51 segments or turns per pole. POWER AND EFFECTIVE VALUES 15 8 poles give 4 cycles per revolution, 500 rev. per min. gives 50 %Q = 8.33 rev. per sec. Thus the frequency is/ = 4 X 8.33 = 33.3 cycles per second. The generated e.m.f. is E = 550 volts, thus by the formula of direct-current generator, E = 4/n, or, 550 = 4 X 0.333 X 51 , = 8.1 megalines per pole. 19. (3) What is the e.m.f. generated in a single turn of a 20-pole alternator running at 200 rev. per min., through a magnetic field of 6.4 megalines per pole? 20 y 200 The frequency is / = V~ ~ = 33 - 3 c y cles - ^ X. OU e = E Q sin r, E Q = 2 irfn3>, $ = 6.4, n = l, f = 0.333. Thus, E Q = 2 TT X 0.333 X 6.4 = 13.4 volts maximum, or e = 13.4 sin 0. 4. POWER AND EFFECTIVE VALUES 20. The power of the continuous e.m.f. E producing con- tinuous current / is P = El. The e.m.f. consumed by resistance r is EI = 7r, thus the power consumed by resistance r is P = 7 2 r. Either EI = E, then, the total power in the circuit is con- sumed by the resistance, or EI < E } then only a part of the power is consumed by the resistance, the remainder by some counter e.m.f., E EI. If an alternating current i = I sin 6 passes through a resist- ance r, the power consumed by the resistance is, i*r = 7 2 r sin 2 = ^r C 1 ~ cos 2 0), & thus varies with twice the frequency of the current, between zero and 7 V. The average power consumed by resistance r is, avg. since avg. (cos) = 0. 16 ELEMENTS OF ELECTRICAL ENGINEERING Thus the alternating current i = IQ since consumes in a resist- ance r the same effect as a continuous current of intensity The value / = 7= is called the effective value of the alter- V2 nating current i = IQ sin 0; since it gives the same effect. ET Analogously E = i is the effective value of the alternating V2 e.m.f., e = E Q sin 6. Since E = 2 irfn$, it follows that J' ; = 4.44 fn& ; is the effective alternating e.m.f. generated in a coil of turns n rotating at a frequency of / (in hundreds of cycles per second) through a magnetic field of megalines of force. This is the formula of the alternating-current generator. 21. The formula of the direct-current generator, E = holds even if the e.m.fs. generated in the individual turns are not sine waves, since it is the average generated e.m.f. The formula of the alternating-current generator, E = V2 *fn$, does not hold if the waves are not sine waves, since the ratios of average to maximum and of maximum to effective e.m.f. are changed. If the variation of magnetic flux is not sinusoidal, the effective generated alternating e.m.f. is, E = 7 \/2 7 is called the form factor of the wave, and depends upon its shape, that is, the distribution of the magnetic flux in the magnetic field. Frequently form factor is defined as the ratio of the effect- ive to the average value. This definition is undesirable since it gives for the sine wave, which is always considered the standard wave, a value differing from one. POWER AND EFFECTIVE VALUES 17 EXAMPLES 22. (1) In a star-connected 20-pole three-phase machine, re- volving at 33.3 cycles or 200 rev. per min., the magnetic flux per pole is 6.4 megalines. The armature contains one slot per pole and phase, and each slot contains 36 conductors. All these conductors are connected in series. What is the effective e.m.f. per circuit, and what the effective e.m.f. between the terminals of the machine? Twenty slots of 36 conductors give 720 conductors, or 360 turns in series. Thus the effective e.m.f. is,. = 4.44 X 0.333 X 360 X 6.4 = 3400 volts per circuit. The e.m.f. between the terminals of a star-connected three- phase machine is the resultant of the e.m.fs. of the two phases, which differ by 60 degrees, and is thus 2 sin 60 = -\/3 times that of one phase, thus, E = = 5900 volts effective. 23. (2) The conductor of the machine has a section of 0.22 sq. cm. and a mean length of 240 cm. per turn. At a resistivity (resistance per unit section and unit length) of copper of p = 1.8 X 10~ 6 , what is the e.m.f. consumed in the machine by the resistance, and what the power consumed at 450 kw. output? 450 kw. output is 150,000 watts per phase or circuit, thus 150 000 the current / = omn = 44.2 amperes effective. The resistance of 360 turns of 240 cm. length, 0.22 sq. cm. section and 1.8 X 10~ 6 resistivity, is 360 X 240 X 1.8 X 10~ 6 r = - -rT^ " = 0.71 ohms per circuit. 44.2 amp. X 0.71 ohms gives 31.5 volts per circuit and (44.2) 2 X 0.71 = 1400 watts per circuit, or a total of 3 X 1400 = 4200 watts loss. 24. (3) What is the self-inductance per wire of a three- phase line of 14 miles length consisting of three wires No. (Id = 0.82 cm.), 45 cm. apart, transmitting the output of this 450 kw. 5900- volt three-phase machine? 18 ELEMENTS OF ELECTRICAL ENGINEERING 450 kw. at 5900 volts gives 44.2 amp. per line. 44.2 amp. effective gives 44.2\/2 = 62.5 amp. maximum. 14 miles = 22,400 m. The magnetic flux produced by / amperes in 1000 m. of a transmission line of 2 wires 45 cm. apart and 0.82 cm. diameter was found in paragraph 1, example 3, as 2 $ = 0.188 X 10 6 /, or $ = 0.094 X 10 6 / for each wire. Thus at 22,300 m. and 62.5 amp. maximum, the flux per wire is $ = 22.3 X 62.5 X 0.094 X 10 6 = 131 megalines. Hence the generated e.m.f., effective value, at 33.3 cycles is, E = A/2 */ $ = 4.44 X 0.333 X 131 = 193 volts per line; the maximum value is, #o = E X \/2 = 273 volts per line; and the instantaneous value, e = E sin (0 - 0i) = 273 sin (0 - 0i) ; or, since = 2 irft = 210 t we have, e = 273 sin 210 (t - h). 25. (4) What is the form factor (a) of the e.m.f. gene- rated in a single conductor of a direct-current machine hav- ing 80 per cent, pole arc and negligible spread of the mag- netic flux at the pole corners, and (6) what is the form fac- tor of the voltage between two collector rings connected to diametrical points of the arm- ature of such a machine? (a) In a conductor during the motion from position A, T? *, T^- shown in Fig. 7, to position FIG. 7. Diagram of bipolar generator. 15, no e.m.f. is generated; from position B to C a constant e.m.f. e is generated, from C to E again no e.m.f., from E to F a constant e.m.f. e, POWER AND EFFECTIVE VALUES 19 and from F to A again zero e.m.f. The e.m.f. wave thus is as shown in Fig. 8. The average e.m.f. is ei = 0.8 e; hence, with this average e.m.f., if it were a sine wave, the maxi- mum e.m.f. would be e z = | ei = 0.4 ire, and the effective e.m.f. would be C D FIG. 8. E.m.f. of a single conductor, direct-current machine 80 per cent, pole arc. The actual square of the e.m.f. is e 2 for 80 per cent, and zero for 20 per cent, of the period, and the average or mean square thus is 0.8 e 2 , and therefore the actual effective value, The form factor 7, or the ratio of the actual effective value e 4 to the effective value e 3 of a sine wave of the same mean value and thus the same magnetic flux, then is e 4 VT6 T = e 3 = ^T = 1.006; that is, practically unity. (6) While the collector leads a, b move from the position F, C, as shown in Fig. 6, to B, E, constant voltage E exists between them, the conductors which leave the field at C being replaced 20 ELEMENTS OF ELECTRICAL ENGINEERING by the conductors entering the field at B. During the motion of the leads a, b from B, E to C, F, the voltage steadily decreases, reverses, and rises again, to E, as the conductors entering the field at E have an e.m.f. opposite to that of the conductors leaving at C. Thus the voltage wave is, as shown by Fig. 9, triangular, with the top cut off for 20 per cent, of the half wave. FIG. 9. E.m.f. between two collector rings connected to diametrical points of the armature of a bipolar machine having 80 per cent, pole arc. Then the average e.m.f. is e 1 = 0.2 E + 2 X = 0.6 E. The maximum value of a sine wave of this average value is e 2 = 2 e i 0-3 irE, and the effective value corresponding thereto is e- 2 0.3 irE 63 = 7= V2 The actual voltage square is E 2 for 20 per cent, of the time, and rising on a parabolic curve from to E 2 during 40 per cent, of the time, as shown in dotted lines in Fig. 9. The area of a parabolic curve is width times one-third of height, or OAE 2 hence, the mean square of voltage is and the actual effective voltage is _, 4 1/280 ~ e, ~ TT V 27 L 25 ' SELF-INDUCTANCE AND MUTUAL INDUCTANCE 21 hence, the form factor is 7 r or, 2.5 per cent, higher than with a sine wave. 5. SELF-INDUCTANCE AND MUTUAL INDUCTANCE 26. The number of inter-linkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in the circuit is called the inductance of the circuit. The number of interlinkages of an electric circuit with the lines of magnetic force of the flux produced by unit current in a second electric circuit is called the mutual inductance of the second upon the first circuit. It is equal to the mutual induc- tance of the first upon the second circuit, as will be seen, and thus is called the mutual inductance between the two circuits. The number of interlinkages of an electric circuit with the lines of magnetic flux produced by unit current in this circuit and not interlinked with a second circuit is called the self- inductance of the circuit. If i = current in a circuit of n turns, = flux produced thereby and interlinked with the circuit, n$ is the total number 9? ^^ of interlinkages, and L = r- the inductance of the circuit. If $ is proportional to the current i and the number of turns n, ni n 2 . , $ = , and L = the inductance. 01 (K (ft is called the reluctance and ni the m.m.f. of the magnetic circuit. In magnetic circuits the reluctance (R has a position similar to that of resistance r in electric circuits. The reluctance (R, and therefore the inductance, is not con- stant in circuits containing magnetic materials, such as iron, etc. If (Ri is the reluctance of a magnetic circuit interlinked with two electric circuits of n\ and n% turns respectively, the flux produced by unit current in the first circuit and interlinked with the second circuit is -- and the mutual inductance of the first (HI upon the second circuit is M = , that is, equal to the Oil 22 ELEMENTS OF ELECTRICAL ENGINEERING mutual inductance of the second circuit upon the first circuit, as stated above. If no flux leaks between the two circuits, that is, if all flux is interlinked with both circuits, and LI = inductance of the first, L 2 = inductance of the second circuit, and M = mutual induc- tance, then M 2 = ZaL 2 . If flux leaks between the two circuits, then M 2 < LiL 2 . In this case the total flux produced by the first circuit con- sists of a part interlinked with the second circuit also, the mu- tual inductance, and a part passing between the two circuits, that is, interlinked with the first circuit only, its self-inductance. 27. Thus, if LI and L 2 are the inductances of the two circuits, and is the total flux produced by unit current in the first n\ HZ and second circuit respectively. T Sf Of the flux --a part is interlinked with the first circuit ni HI only, Si being its self-inductance or leakage inductance, and a part interlinked with the second circuit also, M being the mutual inductance and 1 = + HI HI n 2 Thus, if LI and L 2 = inductance, Si and Sz = self-inductance, M = mutual inductance of two circuits of n and n 2 turns respectively, we have h = Sl + *L Lz = Sz M HI HI n z nz ~ HZ n\ or Li = Si + M L 2 = Sz + - M, HZ HI or M 2 = (Li - Si)(L z - Sz). The practical unit of inductance is 10 9 times the absolute unit or 10 8 times the number of interlinkages per ampere (since 1 amp. = 0.1 unit current), and is called the henry (h); 0.001 of it is called the milhenry (mh.). The number of interlinkages of i amperes in a circuit of SELF-INDUCTANCE AND MUTUAL INDUCTANCE 23 L henry inductance is iL 10 8 lines of force turns, and thus the e.m.f. generated by a change of current di in time dt is e = -j-. L 10 8 absolute units r dt = -T.L volts. at A change of current of 1 amp. per second in the circuit of 1 h. inductance generates 1 volt. EXAMPLES 28. (1) What is the inductance of the field of a 20-pole alternator, if the 20 field spools are connected in series, each spool contains 616 turns, and 6.95 amp. produces 6.4 mega- lines per pole? The total number of turns of all 20 spools is 20 X 616 = 12,320 Each is interlinked with 6.4 X 10 6 lines, thus the total number of interlinkages at 6.95 amp. is 12,320 X 6.4 X 10 6 = 78 X 10 9 . 6.95 amp. = 0.695 absolute units, hence the number of in- terlinkages per unit current, or the inductance, is - 112 X I*- 112 h. 29. (2) What is the mutual inductance between an alter- nating transmission line and a telephone wire carried for 10 miles below and 1.20 m. distant from the one, 1.50 m. distant from the other conductor of the alternating line; and what is the e.m.f. generated in the telephone wire, if the alternating cir- cuit carries 100 amp. at 60 cycles? The mutual inductance between the telephone wire and the electric circuit is the magnetic flux produced by unit current in the telephone wire and interlinked with the alternating circuit, that is, that part of the magnetic flux produced by unit current in the telephone wire, which passes between the dis- tances of 1.20 and 1.50 m. At the distance l x from the telephone wire the length of mag- netic circuit is 2irl z . The magnetizing force / = - if 7 = 24 ELEMENTS OF ELECTRICAL ENGINEERING current in telephone wire in amperes, and the field intensity d the 0.27 H = 0.4 TT/ = , and the flux in the zone dl x is j dl x . l x I = 10 miles = 1610 X 10 3 cm. thus, f 150 0.2// = I i dl x Jl20 * = 322 X 10 3 71ogei|| = 72 7 10 3 ; or, 72 7 10 3 interlinkages, hence, for 7 = 10, or one absolute unit, thus, M = 72 X 10 4 absolute units = 72 X 10~ 5 h. = 0.72 mh. 100 amp. effective or 141.4 amp. maximum or 14.14 abso- lute units of current in the transmission line produces a maximum flux interlinked with the telephone line of 14.14 X 0.72 X 10~ 3 X 10 9 = 10.2 megalines. Thus the e.m.f. generated at 60 cycles is E = 4.44 X 0.6 X 10.2 = 27.3 volts effective. 6. SELF-INDUCTANCE OF CONTINUOUS-CURRENT CIRCUITS 30. Self-inductance makes itself felt in continuous-current circuits only in starting and stopping or, in general, when the current changes in value. Starting of Current. If r = resistance, L = inductance of circuit, E = continuous e.m.-f. impressed upon circuit, i = current in circuit at time t after impressing e.m.f. E, and di the increase of current during time moment dt, then the increase of magnetic interlinkages during time dt is IM, and the e.m.f. generated thereby is r di ei = - L ~di By Lentz's law it is negative, since it is opposite to the im- pressed e.m.f., its cause. Thus the e.m.f. acting in this moment upon the circuit is E + ei = E - L CONTINUOUS-CURRENT CIRCUITS 25 and the current is or, transposing, _ r dt rdt di L i- r the integral of which is . E rt . /. E\ . - L --= log, [i - -) - log, c, where log, c = integration constant. This reduces to - E _i - 7 i = - + ce L at t = 0, i = 0, and thus E -~ = c. Substituting this value, the current is and the e.m.f. of inductance is Att = co , i'o Substituting these values, _ rf = ir E = Ee~ L' r -' i = 0. i = i Q (l - e and Z. The expression u = j is called the "attenuation constant," and its reciprocal, , the "time constant of the circuit." 1 1 The name time constant dates back to the early days of telegraphy, where it was applied to the ratio : , that is, the reciprocal of the attenuation con- stant. This quantity which had gradually come into disuse, again became of importance when investigating transient electric phenomena, and in this work it was found more convenient to denote the value Y as attenuation constant, since this value appears as one term of the more gen- eral constant of the electric circuit ( Y + ~r< ) (Theory and Calculation of Transient Electric Phenomena and Oscillations, Section IV.) 26 ELEMENTS OF ELECTRICAL ENGINEERING Substituted in the foregoing equation this gives and ei = - = - 0.368 E. 31. Stopping of Current. In a circuit of inductance L and E resistance r, let a current IQ = be produced by the impressed e.m.f. E, and this e.m.f. E be withdrawn and the circuit closed through a resistance r\. Let the current be i at the time t after withdrawal of the e.m.f. E and the change of current during time moment dt be di. di is negative, that is, the current decreases. The decrease of magnetic interlinkages during moment dt is Ldi. Thus the e.m.f. generated thereby is T di ei== ~ L di It is negative since di is negative, and e\ must be positive, that is, in the same direction as E, to maintain the current or oppose the decrease of current, its cause. Then the current is e\ L di _ = r + n r + ri dt or, transposing, the integral of which is t = log, i - log, c, where log c = integration constant. r+n This reduces to i = ce L 77* for t = 0, t CONTINUOUS-CURRENT CIRCUITS 27 Substituting this value, the current is E (r + n) t i = -~ L , and the generated e.m.f. is r + r, _fc + ei = t (r + fi) Js c i r p Substituting IQ = , the current is i = lot and the generated e.m.f. is 6l = H(r-f TI) JT-', At * = 0, that is, the generated e.m.f. is increased over the previously impressed e.m.f. in the same ratio as the resistance is increased. When TI = 0, that is, when in withdrawing the impressed e.m.f. E the circuit is short circuited, E _L _';! i = c L = i o z, the current, and _ TL _ !l ei = E L = i re the generated e.m.f. In this case, at t = 0, e\ = E, that is, the e.m.f. does not rise. In the case r\ = < , that is, if in withdrawing the e.m.f. E the circuit is broken, we have t = and ei = , that is, the e.m.f. rises infinitely. The greater r\, the higher is the generated e.m.f. e\, the faster, however, do e\ and i decrease. If n = r, we have at t = 0, 611 = 2E, i = i , and en i* W = | z' Jo that is, the energy stored as magnetism in a circuit of current i Q and inductance L is 2 ' which is independent both of the resistance r of the circuit and the resistance n inserted in breaking the circuit. This energy has to be expended in stopping the current. EXAMPLES 32. (1) In the alternator field in Section 1, Example 4, Sec- tion 2, Example 2, and Section 5, Example 1, how long a time after impressing the required e.m.f. E = 230 volts will it take for the field to reach (a) J/ strength, (b) %Q strength? (2) If 500 volts are impressed upon the field of this alternator, and a non-inductive resistance inserted in series so as to give the required exciting current of 6.95 amp., how long after impressing the e.m.f. E = 500 volts will it take for the field to reach (a) y% strength, (b) %o strength, (c) and what is the resist- ance required in the rheostat? (3) If 500 volts are impressed upon the field of this alter- nator without insertion of resistance, how long will it take for the field to reach full strength? (4) With full field strength, what is the energy stored as magnetism? (1) The resistance of the alternator field is 33.2 ohms (Section 2, Example 2), the inductance 112 h. (Section 5, Example 1), the impressed e.m.f. is E = 230, the final value of current E io = = 6.95 amp. Thus the current at time t is t = * - 6 = 6.95 (1 - e-- 296< ). CONTINUOUS-CURRENT CIRCUITS 29 (a) M strength: i = ~, hence (1 - -- 29 = .5. e -o.296 1 = 0.5, - 0.296 Mog e = log 0.5, t = t = 2.34 seconds. (b) % strength: i = 0.9 * 0j hence (1 - e -- 296 = 0.9, and t = 7.8 seconds. (2) To get io = 6.95 amp., with E = 500 volts, a resist- 500 ance r = ^-f-= = 72 ohms, and thus a rheostat having a resist- o.9o ance of 72 33.2 = 38.8 ohms, is required. We then have i = io (l 2) = 6.95 (1 - -- 643 0. (a) i = ^, after t = 1.08 seconds. & (b) i = 0.9 i , after i = 3.6 seconds. (3) Impressing E = 500 volts upon a circuit of r = 33.2, L = 112, gives = 15.1 (1 - -- 296 0. i = 6.95, or full field strength, gives 6.95 = 15.1 (1 - e -- 296 0. 1 - -- 296 = 0.46 and t = 2.08 seconds. (4) The stored energy is 6.95 2 X 112 ~ = 2720 watt-seconds or joules _ _ = 2000 foot-pounds. (1 joule = 0.736 foot-pounds.) Thus in case (3), where the field reaches full strength in 2.08 2000 seconds, the average power input is - c -^ = 960 foot pounds Z.Oo per second = 1.75 hp. In breaking the field circuit of this alternator, 2000 foot- pounds of energy have to be dissipated in the spark, etc. 33. (5) A coil of resistance r = 0.002 ohm and inductance L = 0.005 mh., carrying current / = 90 amp., is short circuited. 30 ELEMENTS OF ELECTRICAL ENGINEERING (a) What is the equation of the current after short circuit? (6) In what time has the current decreased to 0.1. its initial value? _ L* (a) i = /e L = 90 e- 400 '. (6) i = 0.1 7, c- 400 < = 0.1, after t = 0.00576 second. (6) When short circuiting the coil in Example 5, an e.m.f. E = 1 volt is inserted in the circuit of this coil, in opposite direc- tion to the current. (a) What is equation of the current? (6) After what time does the current become zero? (c) After what time does the current reverse to its initial value in opposite direction? (d) What impressed e.m.f. is required to make the current die out in Hooo second? (e) What impressed e.m.f. E is required to reverse the current in Kooo second? (a) If e.m.f. E is inserted, and at time t the current is denoted by i, we have di ei = L -r, the generated e.m.f. ; Thus, - E + 61 = - E - L j t , the total e.m.f.; and -E + ei E L di ^ i = = -r., the current; r r r dt } Transposing, r , di and integrating, - j- = log, (- + i) - log, c, where log^ c = integration constant. At t = 0, i Substituting, E At t = 0, i = /, thus c = / + -; t E\ - E h 7r "7' Kon ,400 1 p;nn *J*S\J C where t Q is time of one 1 complete period, = -v or by the time angle 6 = 90. FIG. 11. Self-induction effects produced by an alternating sine wave of current. This e.m.f. is called the counter e.m.f. of inductance. It is .'' ' e '*=- L j t = - 2 TT/L/O cos 2 irft. It is shown in dotted line in Fig. 11 as e' 2 . The quantity 2 irfL is called the inductive reactance of the circuit, and denoted by x. It is of the nature of a resistance, and expressed in ohms. If L is given in 10 9 absolute units or henrys, x appears in ohms. The counter e.m.f. of inductance of the current, i = /o sin 2 irft = /o sin 0) of effective value , V"2 IS e' 2 = xI cos 2 irft = xI Q cos 6, having a maximum value of X!Q and an effective value of xh T E, = -- = xl; ALTERNATING-CURRENT CIRCUITS 33 that is, the effective value of the counter e.m.f. of inductance equals the reactance, x, times the effective value of the current, /, and lags 90 time degrees, or a quarter period, behind the current. 35. By the counter e.m.f. of inductance, e'z = xI Q cos 0, which is generated by the change in flux due to the passage of the current i IQ sin through the circuit of reactance x, an equal but opposite e.m.f. e z = xI Q cos is consumed, and thus has to be impressed upon the circuit. This e.m.f. is called the e.m.f. consumed by inductance. It is 90 time degrees, or a quarter period, ahead of the current, and shown in Fig. 11 as a drawn line e 2 . Thus we have to distinguish between counter e.m.f. of induc- tance 90 time degrees lagging, and e.m.f. consumed by inductance 90 time degrees leading. These e.m.fs. stand in the same relation as action and reaction in mechanics. They are shown in Fig. 11 as e' 2 and as e z . The e.m.f. consumed by the resistance r of the circuit is pro- portional to the current, 61 = ri = r/ sin 0, and in phase therewith, that is, reaches its maximum and its zero value at the same time as the current i, as shown by drawn line 61 in Fig. 11. Its effective value is EI = ri. The resistance can also be represented by a (fictitious) counter e.m.f., e\ = r/ sin 0, opposite in phase to the current, shown as e\ in dotted line in Fig. 11. The counter e.m.f. of resistance and the e.m.f. consumed by resistance have the same relation to each other as the counter e.m.f. of inductance and the e.m.f. consumed by inductance or inductive reactance. 36. If an alternating current i = 7 sin of effective value I = ^ exists in a circuit of resistance r and inductance L, that is, of reactance x = 2 irfL, we have to distinguish ; 3 34 ELEMENTS OF ELECTRICAL ENGINEERING E.m.f. consumed by resistance, e\ = r! sin 6, of effective value EI = rl, and in phase with the current. Counter e.m.f. of resistance, e'\ = r/o sin 6, of effective value EI = rl, and in opposition or 180 time degrees displaced from the current. E.m.f. consumed by reactance, e z = X!Q cos 6, of effective value E 2 = xl, and leading the current by 90 time degrees or a quarter period. Counter e.m.f. of reactance, e' z = xlo cos 0, of effective value E'z = xl, and lagging 90 time degrees or a quarter period behind the current. The e.m.fs. consumed by resistance and by reactance are the e.m.fs. which have to be impressed upon the circuit to overcome the counter e.m.fs. of resistance and of reactance. Thus, the total counter e.m.f. of the circuit is e' = e'i + e'z = IQ (r sin 6 + x cos-0), and the total impressed e.m.f., or e.m.f. consumed by the circuit, is e = ei + e z = /o (r sin 6 + x cos 0). Substituting 9S - = tan and VV 2 + r 2 = z, it follows that x = z sin , r = z cos , and we have as the total impressed e.m.f. e = Z!Q sin (0 + ), shown by heavy drawn line e in Fig. 11, and total counter e.m.f. e' = - zI sin (0 + ), shown by heavy dotted line e' in Fig. 11, both of effective value e = zl. For = , e 0, that is, the zero value of e is ahead of the zero value of current by the time angle 0o, or the current lags behind the impressed e.m.f. by the angle . 0o is called the angle of lag of the current, and z = \A" 2 + x 2 the impedance of the circuit, e is called the e.m.f. consumed by impedance, e' the counter e.m.f. of impedance. ALTERNATING-CURRENT CIRCUITS 35 Since Ei = rl is the e.m.f. consumed by resistance, Ez = xl is the e.m.f. consumed by reactance, and E = zl = \/r 2 + x 2 1 is the e.m.f. consumed by impe- dance, we have E = VES + # 2 2 , the total e.m.f. and Ei = E cos , E% = E sin 0o, its components. The tangent of the angle of lag is x 27T/L tan = - = > and the time constant of the circuit is L _ tan &o r = ~27f" The total e.m.f., e, impressed upon the circuit consists of two components, one, e\ t in phase with the current, the other one, e 2 , in quadrature with the current. Their effective values are E, E cos 0) E sin . EXAMPLES 37. (1) What is the reactance per wire of a transmission line of length Z, if l d = diameter of the wire, 1 8 = spacing of the wires, and/ = frequency? If / = current, in absolute units, in one wire of the trans- mission line, the m.m.f. is I; thus the magnetizing force in a zone dl x at distance l x from center of wire (Fig. 12) is / = 7 Z TTlx and the field intensity in this zone is H = 4 irf = 2 y Thus L x the magnetic flux in this zone is d * . H ldli m hence, the total magnetic flux between the wire and the return wire is L XI* d* = $ | CfcSF = ^.f6| -y = 2 1 1 IQge -j > LX I'd 2 2 neglecting the flux inside the transmission wire. 36 ELEMENTS OF ELECTRICAL ENGINEERING The inductance is L = -y = 2 I log e -y^ absolute units -/ 'd 2 Z log e s 1(T 9 h., I'd 21 s and the reactance x = 2 irfL = 4 irfl log e -y-, in absolute units; or x = 4 7T/7 log e -y^ 10~ 9 , in ohms. 'd 38. (2) The voltage at the receiving end of a 33.3-cycle three-phase transmission line 14 miles in length shall be 5500 FIG. 12. Diagram for calculation of inductance between two parallel conductors. between the lines. The line consists of three wires, No. B. & S. (l d = 0.82 cm.), 18 in. (45 cm.) apart, of resistivity p = 1.8 X 10- 6 . (a) What is the resistance, the reactance, and the impedance per line, and the voltage consumed thereby at 44 amp. ? (6) What is the generator voltage between lines at 44 amp. to a non-inductive load? (c) What is the generator voltage between lines at 44 amp. to a load circuit of 45 degrees lag? (d) What is the generator voltage between lines at 44 amp. to a load circuit of 45 degrees lead? Here I = 14 miles = 14 X 1.6 X 10 5 = 2.23 X 10 6 cm. l d = 0.82 cm. Hence the cross section, A = 0.528 sq. cm. ALTERNATING-CURRENT CIRCUITS 37 Z 1.8 X 10- 6 X 2.23 X 10 6 (a) Resistance per line, r = p - = = 7.60 ohms. 2L Reactance per line, x = 4 irfl log, j- X 10~ 9 = ^ X 33.3 X 2.23 X 10 6 X log, 110 X 10~ 9 = 4.35 ohms. The impedance per line, z = \/r 2 -f- x' 2 = 8.76 ohms. Thus if I = 44 amp. per line, the e.m.f. consumed by resistance is EI = rl = 334 volts, the e.m.f. consumed by reactance is E z = xl = 192 volts, and the e.m.f. consumed by impedance is E 3 = zl = 385 volts. (b) 5500 volts between lines at receiving circuit give -j= = v 3 3170 volts between line and neutral or zero point (Fig. 13), or per line, corresponding to a maxi- mum voltage of 3170 A/2 = 4500 volts. 44 amp. effective per line gives a maxi- mum value of 44 -\/2 = 62 amp. Denoting the current by i = 62 sin 0, the voltage per line at the receiv- ing end with non-inductive load is e = 4500 sin 0. The e.m.f. consumed by resistance, in phase with the current, of effective JT IG 13 _ Voltage diagram for value 334, and maximum value 334 a three-phase circuit. \/2 = 472, is ei = 472 sin 0. The e.m.f. consumed by reactance, 90 time degrees ahead of the current, of effective value 192, and maximum value 192 -\/2 = 272, is e 2 = 272 cos 0. Thus the total voltage required per line at the generator end of the line is e Q = e + e l + e 2 = (4500 + 472) sin + 272 cos = 4972 sin + 272 cos 0. 272 Denoting . n _ = tan , we have tan 0o 272 C S " = 4980 1 _ 4972 ~ 4980- 38 ELEMENTS OF ELECTRICAL ENGINEERING Hence, BQ = 4980 (sin cos 0o + cos 6 sin ) = 4980 sin (0 + ). Thus 0o is the lag of the current behind the e.m.f. at the generator end of the line, = 3.2 time degrees, and 4980 the 4980 maximum voltage per line at the generator end; thus E Q = = 3520, the effective voltage per line, and 3520 \/3 = 6100, the effective voltage between the lines at the generator. (c) If the current i = 62 sin lags in time 45 degrees behind the e.m.f. at the receiving end of the line, this e.m.f. is expressed by e = 4500 sin (0 + 45) = 3170 (sin + cos 0); that is, it leads the current by 45 time degrees, or is zero at = 45 time degrees. The e.m.f. consumed by resistance and by reactance being the same as in (6), the generator voltage per line is o = e -f 6l -f e 2 = 3642 sin + 3442 cos 0. 3442 Denoting QA/fo = tan , we have OO4Z e = 5011 sin (0 -f- ). Thus , the angle of lag of the current- behind the gen- erator e.m.f., is 43 degrees, and 5011 the maximum voltage; hence 3550 the effective voltage per line, and 3550 -\/3 = 6160 the effective voltage between lines at the generator. (d) If the current i = 62 sin leads the e.m.f. by 45 degrees, the e.m.f. at the receiving end is e = 4500 sin (0 - 45) = 3180 (sin - cos 0). Thus at the generator end Q = e -f 6l + e2 = 3652 sin - 2908 cos 0. 2908 Denoting = tan ^o, it is e = 4670 sin (0 - ). Thus , the time angle of lead at the generator, is 39 degrees, and 4654 the maximum voltage; hence 3290 the effective vol- tage per line and 5710 the effective voltage between lines at the generator. POWER IN ALTERNATING-CURRENT CIRCUITS 39 8. POWER IN ALTERNATING-CURRENT CIRCUITS of effective value I = 7=-, in a circuit of resistance r and reac- V2 39. The power consumed by alternating current i = I sin 0, effective value I tance x = 2 nfL, is p = ei, where e = z! Q sin (0 + ) is the impressed e.m.f., consisting of the components ei = r/ sin 0, the e.m.f. consumed by resistance and 62 = x! Q cos 0, the e.m.f. consumed by reactance. z = \/r 2 + x 2 is the impedance and tan = the phase angle of the circuit; thus the power is p = z/o 2 sin sin (0 + ) = ^- (OS - COS (20+ )) = zP (cos - cos (20 + )). Since the average cos (20 + ) = zero, the average power is P = zP cos = rP = EJ-, that is, the power in the circuit is that consumed by the resistance, and independent of the reactance. Reactance or self-inductance consumes no power, and the e.m.f. of self-inductance is a wattless or reactive e.m.f., while the e.m.f. of resistance is a power or active e.m.f. The wattless e.m.f. is in quadrature, the power e.m.f. in phase with the current. In general, if = angle of time-phase displacement between the resultant e.m.f. and the resultant current of the circuit, / = current, E = impressed e.m.f., consisting of two com- ponents, one, EI = E cos 0, in phase with the current, the other, 1 2 = E sin 0, in quadrature with the current, the power in the circuit is IEi = IE cos 0, and the e.m.f. in phase with the current Ei = E cos is a power e.m.f., the e.m.f. in quadrature with the current E 2 = E sin a wattless or reactive e.m.f. 40 ELEMENTS OF ELECTRICAL ENGINEERING 40. Thus we have to distinguish power e.m.f. and wattless or reactive e.m.f., or power component of e.m.f., in phase with the current and wattless or reactive component of e.m.f., in quadra- ture with the current. Any e.m.f. can be considered as consisting of two components, one, the power component, e\, in phase with the current, and the other, the reactive component, e z , in quadrature with the current. The sum of instantaneous values of the two compo- nents is the total e.m.f. e = ei + e* If E } EI, Ez are the respective effective values, we have E = Ei* + E 2 2 , since EI = E cos &, E 2 = E sin 6, where = phase angle between current and e.m.f. Analogously, a current I due to an impressed e.m.f. E with a time-phase angle can be considered as consisting of two component currents, 1 1 = I cos 8, the active or power component of the current, and J 2 = / sin 0, the wattless or reactive component of the current. The sum of instantaneous values of the power and reactive components of the current equals the instantaneous value of the total current, ii + iz = i, while their effective values have the relation i = V77+77 2 . Thus an alternating current can be resolved in two com- ponents, the power component, in phase with the e.m.f., and the wattless or reactive component, in quadrature with the e.m.f. An alternating e.m.f. can be resolved in two components: the power component, in phase with the current, and the watt- less or reactive component, in quadrature with the current. The power in the circuit is the current times the e.m.f. times the cosine of the time-phase angle, or is the power component of the current times the total e.m.f., or the power component of the e.m.f. times the total current. VECTOR DIAGRAMS 41 EXAMPLES 41. (1) What is the power received over the transmission line in Section 7, Example 2, the power lost in the line, the power put into the line, and the efficiency of transmission with non- inductive load, with 45-time-degree lagging load and 45-degree leading load? The power received per line with non-inductive load is P = El = 3170 X 44 = 139 kw. With a load of 45 degrees phase displacement, P = El cos 45 = 98 kw. The power lost per line PI = PR = 44 2 X 7.6 = 14.7 kw. Thus the input into the line P = P + PI = 151.7 kw. at non-inductive load, and = 111.7 kw. at load of 45 degrees phase displacement. The efficiency with non-inductive load is P 14 7 Po = l - 15T7 = )0 - 3 p and with a load of 45 degrees phase displacement is P 14.7 ^- = 1 111 -, = 86.8 per cent. L Q 111./ The total output is 3 P = 411 kw. and 291 kw., respectively. The total input 3 P = 451.1 kw. and 335.1 kw., respectively. 9. VECTOR DIAGRAMS 42. The best way of graphically representing alternating-cur- rent phenomena is by a vector diagram. The most frequently used vector diagram is the crank diagram. In this, sine waves of alternating currents, voltages, etc., are represented as projec- tions of a revolving vector on the horizontal. That is, a vector equal in length to the maximum value of the alternating wave is assumed to revolve at uniform speed so as to make one complete revolution per period, and the projections of this revolving vec- tor upon the horizontal then represent the instantaneous values of the wave. Let, for instance, 01 represent in length the maximum value of current i = I cos (6 ). Assume then a vector, 07, to revolve, left-handed or in positive direction, so that it makes a 42 ELEMENTS OF ELECTRICAL ENGINEERING complete revolution during each cycle or period. If then at a certain moment of time this vector stands in position OIi (Fig. 14), the projection, OA^' of Oh on OA represents the instan- taneous value of the current at this moment. At a later moment 07 has moved farther, to 0/ 2 , and the projection, OA Z , of 07 2 on OA is the instantaneous value. The diagram thus shows the instantaneous condition of the sine waves. Each sine wave FIG. 14. Crank diagram showing instantaneous values. FIG. 15. Crank diagram of an e.m.f. and current. reaches the maximum at the moment when its revolving vector, 01, passes the horizontal, and reaches zero when its revolving vector passes the vertical. If Fig. 15 represents the crank diagram of a voltage OE, and a current^O/, and if angle AOE^AOI, this means that the current 01 is behind the voltage OE, passes during the revolu- tion the zero line or line of maximum intensity, OA, later than the voltage; that is, the current lags behind the voltage. In the vector diagram, the first quantity therefore can be put in any position. For_instance, the current 01, in Fig. 15, could be drawn in position 01, Fig. 16. The voltage then being ahead VECTOR DIAGRAMS 43 of the current by angle EOI = would come into the position OE, Fig. 16. This vector diagram then shows graphically, by the projections of the vectors on the horizontal, the instantaneous values of the alternating waves at one moment of time. At any other moment FIG. 16. Crank diagram. of time, the instantaneous values would be the projections of the vectors on another radius, corresponding to the other time. The angles between the vector representation are the phase differ- ences between the vectors, and the angles each vector makes with the horizontal may be called its phase. The horizontal then FIG. 17. Vector diagram of two e.m.f.'s acting in the same circuit. would be of phase zero. The phase of the first vector may be chosen at random; all other phases are determined thereby. In this representation, the phase of an alternating wave is given by the time when its maximum value passes the horizontal. 44 ELEMENTS OF ELECTRICAL ENGINEERING Two voltages, e\ and e 2 , acting in the same circuit, give a resultant voltage e equal to the sum of their instantaneous values. Graphically, voltages ei and e% are represented in intensity and in phase by two revolving vectors, OEi and OE Z , Fig. 17. The instantaneous values are the projections Oei, Oe 2 of OEi and OE Z upon the horizontal. Since the sum of the projections of the sides of a parallelogram is equal to thejDrojection of the diagonal,jthe sum of the projec- tions Oei and Oe z equals the projection Oe of OE, the diagonal of the parallelogram with OEi and OE Z as sides, and OE is thus the resultant e.m.f . ; that is, graphically alternating sine waves of voltage, current, etc., are combined and resolved by the parallelo- gram or polygon of sine waves. FIG. 18. Vector diagram. 43. The sine wave of alternating current i = I sin is repre- sented by a vector equal in length, 01 , to the maximum value 7 of the wave, and located so that at time zero 0=0, its projec- tion on the horizontal, is zero, and at times > 0, but < TT, the projection is positive. Thus this vector 0/ is the negative vertical, as shown in Fig. 18. The voltage consumed by inductance, e z = x! cos 0, is repre- sented by a vector OE Z equal in length to x! Q , and located so that at = 0, its projection on the horizontal is a maximum. That is, it is the zero vector OE 2 in Fig. 18. Analogously, the counter e.m.f. of self-inductance E' 2 is represented by vector OE' Z on the negative horizontal of Fig. 18; the voltage consumed by the resistance r, e\ e! Q sin 0, is represented by vector OEi equal to r/ , and located on the nega- VECTOR DIAGRAMS 45 tive vertical, and the counter e.m.f. of resistance by vector OE'i on the positive vertical. The counter e.m.f. of impedance: (r/o sin + x! Q cos 0) - ?J n sin (ft -\- fi} sin (6 + ) then is represented graphically as the resultant, by the parallelo- gram of sine waves of OE\ and OE' 2} that is, by a vector OE', equal in length to z! , and of phase 90 + . The voltage consumed by impedance, or the impressed voltage, is represented by the vector OE, equal and opposite in direction to the vector OE' . This vector is the resultant of OEi and OE 2 and has the phase 90, or (90 ), as shown in Fig. 18. An alternating wave is thus determined by the length and direc- tion of its vector. The length is the maximum value, intensity or amplitude of the wave; the direction is the phase of its maximum value, usually called the phase of the wave. 44. As phase of the first quantity considered, as in the above instance the current, any direction can be chosen. The further quantities are determined thereby in direction or phase. The zero vector OA is generally chosen for the most frequently used quantity or reference quantity, as for the current, if a num- ber of e.m.fs. are considered in a circuit of the same current, or for the e.m.f., if a number of currents are produced by the same e.m.f., or for the generated e.m.f. in apparatus such as transform- ers and induction motors, synchronous apparatus, etc. With the current as zero vector, all horizontal components of e.m.f. are power components, all vertical components are reac- tive components. With the e.m.f. as zero vector, all horizontal components of current are power components, all vertical components of current are reactive components. By measurement from the vector diagram numerical values can hardly ever be derived with sufficient accuracy, since the magnitudes of the different quantities used in the same diagram are usually by far too different, and the vector diagram is there- fore useful only as basis for trigonometrical or other calculation, and to give an insight into the mutual relation of the different quantities, and even then great care has to be taken to distinguish between the two equal but opposite vectors, counter e.m.f. and e.m.f. consumed by the counter e.m.f., as explained before. 46 ELEMENTS OF ELECTRICAL ENGINEERING EXAMPLES 45. In a three-phase long-distance transmission line, the vol- tage between lines at the receiving end shall be 5000 at no load, 5500 at full load of 44 amp. power component, and propor- tional at intermediary values of the power component of the current; that is, the voltage at the receiving end shall increase proportional to the load. At three-quarters load the current shall be in phase with the e.m.f. at the receiving end. The generator excitation, however, and thus the (nominal) generated FIG. 19. Vector diagram of e.m.f. and current in transmission line. Cur- rent leading. e.m.f. of the generator shall be maintained constant at all loads, and the voltage regulation effected by producing lagging or leading currents with a synchronous motor in the receiving cir- cuit. The line has a resistance r x = 7.6 ohms and a reactance Xi = 4.35 ohms per wire, the generator is star connected, the resistance per circuit being r 2 = 0.71, and the (synchronous) reactance is x 2 = 25 ohms. ^ What must be the wattless or re- active component of the current, and therefore the total current and its phase relation at no load, one-quarter load, one-half load, three-quarters load, and full load, and what will be the terminal voltage of the generator under these conditions? The total resistance of the line and generator is r = TI + r 2 = 8.31 ohms; the total reactance, x = Xi + # 2 = 29.35 ohms. Let, in the polar diagram, Fig. 19 or 20, OE = E represent the voltage at the receiving end of the line, OIi = I\ the power component of the current corresponding to the load, in phase with OE, and 0/2 = Iz the reactive component of the current in quadrature with OE, shown leading in Fig. 19, lagging in Fig. 20. We then have total current / = 01. VECTOR DIAGRAMS 47 Thus the e.m.f. consumed by resist ance_,j9'i = rl, is in phase with 7,the e.m.f. consumed by reactance, OEz = xl, is 90 degrees ahead of /, and their resultant is OE 3 , the e.m.f. consumed by impedance. OW 3 combined with 0#, the receiver voltage, gives the genera- tor voltage OE . FIG. 20. Vector diagram of e.m.f. and current in transmission line. Cur- rent lagging. Resolving all e.m.fs. and currents into components in phase and in quadrature with the received voltage E, we have Current E.m.f. at receiving end of line, E = E.m.f. consumed by resistance, EI = E.m.f. consumed by reactance, E 2 = Thus total e.m.f. or generator voltage, E = E + E! + E 2 = E + Herein the reactive lagging component of current is assumed as positive, the leading as negative. The generator e.m.f. thus consists of two components, which give the resultant value Phase component Quadrature component /I ~/2 E r/i -r/ 2 xl z + Z/1 + xI xl l - r/ E, = V(E + rh + xI,Y + (xh - r/ 2 ) 2 ; substituting numerical values, this becomes + 8.31 At three-quarters load, 5375 + 29.35 7 2 ) 2 + (29.35 A - 8.31 7 2 ) 2 . E = 3090 volts per circuit, 48 ELEMENTS OF ELECTRICAL ENGINEERING /i = 33, 7 2 = 0, thus Eo = \/(3090 + 8.31 X 33) 2 + (29.35 X 33) 2 = 3520 volts per line or 3520 X \/3 = 6100 volts between lines as (nominal) generated e.m.f. of generator. Substituting these values, we have 3520 = \/(E + 8.31 7i + 29.35 7 2 ) 2 + (8.31 1 2 - 29.35 /O 2 . The voltage between the lines at the receiving end shall be: No U M H Full load load load load load Voltage between lines, 5000 5125 5250 5375 5500 Thus, voltage per line -5- \/3, # = 2880 2950 3020 3090 3160 The power components of current per line, I I = 11 22 33 44 Herefrom we get by substituting in the above equation Reactive component of d lo t d lo ^ d ^ ^d current, 7 2 = -21.6 -16.2 -9.2 +9.7 hence, the total current, + / 2 2 = 21.6 19.6 23.9 33.0 45.05 and the power factor, ^ = cos = 56.0 92.0 100.0 97.7 the lag of the current, = 90 61 23 -11.5 the generator terminal voltage per line is E' = V(E + rj, = V(E + 7.6 A + 4.35 7 2 ) 2 + (4.35 I I - 7.6 7 2 ) 2 thus: No \i M H. Full load load load load load Per line, E'_ = 2980 3106 3228 3344 3463 Between lines, E' V3 = 5200 5400 5600 5800 6000 Therefore at constant excitation the generator voltage rises with the load, and is approximately proportional thereto. 10. HYSTERESIS AND EFFECTIVE RESISTANCE 46. If an alternating current 01 = I, in Fig. 21, exists in a circuit of reactance x = 2 irfL and of negligible resistance, the HYSTERESIS AND EFFECTIVE RESISTANCE 49 magnetic flux produced by the current, 0$ = $, is in phase with the current, and the e.m.f. generated by this flux, or counter e.m.f. of self-inductance, OE'" = E'" = xl, lags 90 degrees be- hind the current. The e.m.f. consumed by self-inductance or impressed e.m.f. OE" = E" = xl is thus 90 degrees ahead of the current. Inversely, if the e.m.f. OE" = E" is impressed upon a circuit of reactance x = 2 irfL and of negligible resistance, the current E" 01 = I = - - lags 90 degrees behind the impressed e.m.f. x This current' is called the exciting or magnetizing current of the magnetic circuit, and is wattless. ' If the magnetic circuit contains iron or other magnetic mate- rial, energy is consumed in the magnetic circuit by a frictional resistance of the material against a change of magnetism, which is called molecular magnetic friction. If the alternating current is the only avail- able source of energy in the magnetic cir- cuit, the expenditure of energy by molec- ular magnetic friction appears as a lag of the magnetism behind the m.m.f. of the Q| r >i current, that is, as magnetic hysteresis, and can be measured thereby. Magnetic hysteresis is, however, a dis- tinctly different phenomenon from molec- ular magnetic friction, and can be more or less eliminated, as for instance by me- chanical vibration, or can be increased, without changing the molecular magnetic friction. 47. In consequence of magnetic hysteresis, if an alternating e.m.f. OE" = E" is im- pressed upon a circuit of negligible resistance, the exciting current, or current producing the magnetism, in this circuit is not a wattless current, or current of 90 degrees lag, as in Fig. 21, but lags less than 90 degrees, by an angle 90 a, as shown by OI = I in Fig. 22. Since the magnetism 0$ = $ is in quadrature with the e.m.f. E" due to it, angle a is the phase difference between the magnet- ism and the m.m.f., or the lead of the m.m.f., that is, the exciting 4 FIG. 21. Phase re- lations of magnetizing current, flux and self- inductive e.m.f. 50 ELEMENTS OF ELECTRICAL ENGINEERING current, before the magnetism. It is called the angle of hysteretic lead. In this case the exciting current 01 = I can be resolved in two components: the magnetizing current 01 2 1 2, in phase with the magnetism 0$ = $, that is, in quadrature with the e.m.f. OE" = E"j and thus wattless, and the magnetic power component of the current or the hysteresis current OIi = Ii, in phase jvvith the e.m.f. OE" = E", or in quadrature with the magnetism 0$ = $. Magnetizing current and hysteresis current are the two com- ponents of the exciting current. FIG. 22. Angle of hysteretic lead. FIG. 23. Effect of resistance on phase relation of impressed e.m.f. in a hysteresisless circuit. If the circuit contains besides the reactance x = 2 wfL, a re- sistance r, the e.m.f. OE" = E" in the preceding Figs. 21 and 22 is not the impressed e.m.f., but the e.m.f. consumed by self- inductance or reactance, and has to be combined, Figs. 23 and 24, with the e.m.f. consumed by the resistance, OE' = E' = Ir, to get the impressed e.m.f. OE = E. Due to the hysteretic lead a, the lag of the current is less in Figs. 22 and 24, a circuit expending energy in molecular mag- netic friction, than in Figs. 21 and 23, a hysteresisless circuit. As seen in Fig. 24, in a circuit whose ohmic resistance is not negligible, the hysteresis current and the magnetizing current are not in phase and in quadrature respectively with the im- pressed e.m.f., but with the counter e.m.f. of inductance or e.m.f. consumed by inductance. Obviously the magnetizing current is not quite wattless, since HYSTERESIS AND EFFECTIVE RESISTANCE 51 energy is consumed by this current in the ohmic resistance of the circuit. Resolving, in Fig. 25, the impressed e.m.f. OE = E into two components, OEi = EI in phase, and OE 2 = E 2 in quadrature with the current 01 = I, the power component of the e.m.f., EI, is greater than E r = Ir, and the reactive component E 2 is less than E" OE, Ix. FIG. 24. Effect of resistance on phase relation of impressed e.m.f. in a circuit having hys- teresis. FIG. 25. Impressed e.m.f. resolved into components in phase and in quadrature with the exciting current. The value r' ance, and the value x' -r EI power e.m.f. . ... . . . . -=- = *- - is called the effective resist- I total current E 2 wattless e.m.f. I is called the ap- total current parent or effective reactance of the circuit. 48. Due to the loss of energy by hysteresis (eddy currents, etc.), the effective resistance differs from, and is greater than, the ohmic resistance, and the apparent reactance is less than the true or inductive reactance. The loss of energy by molecular magnetic friction per cubic centimeter and cycle of magnetism is approximately W = r}B^, where B = the magnetic flux density, in lines per sq. cm. W = energy, in absolute units or ergs per cycle (= 10~ 7 watt-seconds or joules), and t\ is called the coef- ficient of hysteresis. 52 ELEMENTS OF ELECTRICAL ENGINEERING In soft annealed sheet iron or sheet steel and in silicon steel, rj varies from 0.60 X 10~ 3 to 2.5 X 10~ 3 , and can in average, for good material, be assumed as 1.5 X 10~ 3 . The loss of power in the volume, V, at flux density B and frequency /, is thus P = VfoB 1 ' 6 X 10" 7 , in watts, and, if / = the exciting current, the hysteretic effective resist- ance is P B 1 ' 6 r" =J-* = VfrW-^' If the flux density, B, is proportional to the current, /, sub- stituting for B, and introducing the constant k, we have r n V ~ 'PA' that is, the effective hysteretic resistance is inversely propor- tional to the 0.4 power of the current, and directly proportional to the frequency. 49. Besides hysteresis, eddy or Foucault currents contribute to the effective resistance. Since at constant frequency the Foucault currents are pro- portional to the magnetism producing them, and thus approxi- mately proportional to the current, the loss of power by Foucault currents is proportional to the square of the current, the same as the ohmic loss, that is, the effective resistance due to Foucault currents is approximately constant at constant frequency, while that of hysteresis decreases slowly with the current. Since the Foucault currents are proportional to the frequency, their effective resistance varies with the square of the frequency, while that of hysteresis varies only proportionally to the frequency. The total effective resistance of an alternating-current circuit increases with the frequency, but is approximately constant, within a limited range, at constant frequency, decreasing some- what with the increase of magnetism. EXAMPLES 50. A reactive coil shall give 100 volts e.m.f. of self-inductance at 10 amp. and 60 cycles. The electric circuit consists of 200 turns (No. 8 B. & S.) (= 0.013 sq. in.) of 16 in. mean length of turn. The magnetic circuit has a section of 6 sq. in. and a HYSTERESIS AND EFFECTIVE RESISTANCE 53 mean length of 18 in. of iron of hysteresis coefficient rj = 2.5 X 1CT 3 . An air gap is interposed in the magnetic circuit, of a section of 10 sq. in. (allowing for spread), to get the desired reactance. How long must the air gap be, and what is the resistance, the reactance, the effective resistance, the effective impedance, and the power-factor of the reactive coil? The coil contains 200 turns each 16 in. in length and 0.013 sq. in. in cross section. Taking the resistivity of copper as 1.8 X 10~ 6 , the resistance is 1.8 X 10~ 6 X 200 X 16 0.013 X 2.54" im > where 2.54 is the factor for converting inches to centimeters. (1 inch = 2.54 cm.) Writing E = 100 volts generated, / = 60 cycles per second, and n = 200 turns, the maximum magnetic flux is given by E = 4.44 fn$; or, 100 = 4.44 X 0.6 X 200$, and 3> = 0.188 megaline. This gives in an air gap of 10 sq. in. a maximum density B = 18,800 lines per sq. in., or 2920 lines per sq. cm. Ten amperes in 200 turns give 2000 ampere-turns effective or F = 2830 ampere-turns maximum. Neglecting the ampere-turns required by the iron part of the magnetic circuit as relatively very small, 2830 ampere-turns have to be consumed by the air gap of density B = 2920. Since D 4?rF loT the length of the air gap has to be 47TX2830 To* =: ToxWo == L22 cm " or ' 48 ln ' With a cross section of 6 sq. in. and a mean length of 18 in., the volume of the iron is 108 cu. in., or 1770 cu. cm. OOO The density in the iron, BI = -- g -- = 31,330 lines per sq. in., or 4850 lines per sq. cm. With an hysteresis coefficient 77 = 2.5 X 10~ 3 , and density BI = 4850, the loss of energy per cycle per cubic centimeter is W = i/fii 1 - 6 = 2.5 X 10- 3 X 4850 1 - 6 = 1980 ergs, 54 ELEMENTS OF ELECTRICAL ENGINEERING and the hysteresis loss at/ = 60 cycles and the volume V = 1770 is thus P = 60 X 1770 X 1980 ergs per sec. = 21.0 watts, which at 10 amp. represent an effective hysteretic resistance, 21.0 r 2 = -y~j- 0.21 ohm. Hence the total effective resistance of the reactive coil is r = n + r 2 = 0.175 + 0.21 = 0.385 ohm the effective reactance is 777 x = ~j = 10 ohms; the impedance is z = 10.01 ohms; the power-factor is T cos - = 3.8 per cent.; z the total apparent power of the reactive coil is I 2 z = 1001 volt-amperes, and the loss of power, Pr = 38 watts. 11. CAPACITY AND CONDENSERS 51. The charge of an electric condenser is proportional to the impressed voltage, that is, potential difference at its terminals, and to its capacity. A condenser is said to have unit capacity if unit current exist- ing for one second produces unit difference of potential at its terminals. The practical unit of capacity is that of a condenser in which 1 amp. during one second produces 1 volt difference of potential. The practical unit of capacity equals 10~ 9 absolute units. It is called a farad. One farad is an extremely large capacity, and therefore one millionth of one farad, called microfarad, mf., is commonly used. If an alternating e.m.f. is impressed upon a condenser, the charge of the condenser varies proportionally to the e.m.f., and CAPACITY AND CONDENSERS 55 thus there is current to the condenser during rising and from the condenser during decreasing e.m.f., as shown in Fig. 26. That is, the current consumed by the condenser leads the impressed e.m.f. by 90 time degrees, or a quarter of a period. Denoting / as frequency and E as effective alternating e.m.f. impressed upon a condenser of C'mf. capacity, the condenser is charged and discharged twice during each cycle, and the time of one complete charge or discharge is therefore j^- Since E \/2 is the maximum voltage impressed upon the con- denser, an average of CE \/2 10~ 6 amp. would have to exist during one second to charge the condenser to this voltage, and FIG. 26. Charging current of a condenser across which an alternating e.m.f. is impressed. to charge it in j^ seconds an average current of 4 fCE \/2 10~ 6 amp. is required. effective current TT Since average current 2\/2' the effective current is I = 2-irfCE 10~ 6 ; that is, at an impressed e.m.f. of E effective volts and frequency /, a condenser of C mf. capacity consumes a current of 1 = 2 irfCE 10~ 6 amp. effective, which current leads the terminal voltage by 90 degrees or a quarter period. Transposing, the e.m.f. of the condenser is 10 6 / 10 6 The value z = fn is called the condensive reactance of the ^ 7T/C condenser. 56 ELEMENTS OF ELECTRICAL ENGINEERING Due to the energy loss in the condenser by dielectric hysteresis, the current leads the e.m.f. by somewhat less than 90 time de- grees, and can be resolved into a wattless charging current and a dielectric hysteresis current, which latter, however, is generally so small as to be negligible, though in underground cables of poor quality, it may reach as high as 50 per cent, or more of the charging or wattless current of the condenser. 52. The capacity of one wire of a transmission line is i.nxio- 6 x/ . C = - ~-i - , in mf., where Id = diameter of wire, cm.; 1 8 distance of wire from return wire, cm.; I = length of wire, cm., and 1.11 X 10~ 6 = reduction coefficient from electrostatic units to mf . The logarithm is the natural logarithm; thus in common loga- rithms, since loge a = 2.303 logio a, the capacity is 0.24 X 10~ 6 X I i ^ t>s logio -7- I'd . , , in mf . The derivation of this equation must be omitted here. The charging current of a line wire is thus 1 = 2 7T/CE 10~ 6 , where / = the frequency, in cycles per second, E = the difference of potential, effective, between the line and the neutral (E y^ line voltage in a single-phase, or four-wire quarter-phase sys- tem, -i=. line voltage, or Y voltage, in a three-phase system). V 3 EXAMPLES 53. In the transmission line discussed in the examples in 37, 38, 41 and 45, what is the charging current of the line at 6000 volts between lines, at 33.3 cycles? How many volt-amperes does it represent, and what percentage of the full-load current of 44 amp. is it? The length of the line is, per wire, I = 2.23 X 10 6 cm. The distance between wires, l s = 45 cm. The diameter of transmission wire, Id = 0.82 cm. Thus the capacity, per wire, is C = - . 0.26 mf. 1 < t s lo glo - IMPEDANCE OF TRANSMISSION LINES 57 The frequency is / = 33.3, The voltage between lines, 6000. Thus per line, or between line and neutral point, E = = 3460 volts; hence, the charging current per line is Jo = 2 irfCE 10 ~ 6 = 0.19 amp., or 0.43 per cent, of full-load current; that is, negligible in its influence on the transmission voltage. The volt-ampere input of the transmission is, 3 I Q E = 2000 = 2.0 kv-amp. 12. IMPEDANCE OF TRANSMISSION LINES 54. Let r = resistance; x = 2 irfL = the reactance of a trans- mission line; E = the alternating e.m.f. impressed upon the line; I = the line current; E = the e.m.f. at receiving end of the line, and 6 = the angle of lag of current 7 behind e.m.f. E. B < thus denotes leading, > lagging current, and 6 = a non-in- ductive receiver circuit. The capacity of the transmission line shall be considered as negligible. FIG. 27. Vector diagram ,1 i f , v i. of current and e.m.fs. in a _Assummg the phase of the current transmission line assuming QI = / as zero in the polar diagram, zero capacity. Fig. 27, the e.m.f. E is represented by vector OE, ahead of 07 by angle 0. The e.m.f. consumed by re- sistance r is OEi = Ei = Ir in phase with the current, and the e.m.f. consumed by reactance x is OE% = E z = Ix, 90 time de- grees ahead of the current; thus the total e.m.f. consumed by the line, or e.m.f. consumed by impedance, is the resultant OE S of and O#2, jind is E 3 = Iz. Combining OEz and OE gives OE Q , the e.m.f. impressed upon the line. 58 ELEMENTS OF ELECTRICAL ENGINEERING Denoting tan 0i = - the time angle of lag of the line impe- dance, it is, trigonometrically, Since OE 2 = OE 2 + EE Q 2 - 2 OE X EE Q cos ~EEo = OE* = Iz, OEE Q = 180 - 0i + 6, FIG. 28. Locus of the generator and receiver e.m.fs. in a transmission line with varying load phase angle. E 2 = E 2 + I 2 z 2 + 2 EIz cos (0! - 6) = (E + Iz) 2 - 4 #/z sin 2 ^-^, we have and E = \I(E -f- Iz) 2 4 EIz sin 2 -^= , and the drop of voltage in the line, EQ - E = \ (E + Iz} 2 - 4 EIz sin 2 -^ E. IMPEDANCE OF TRANSMISSION LINES 59 65. That is, the voltage E Q required at the sending end of a line of resistance r and reactance x, delivering current / at vol- tage E } and the voltage drop in the line, do not depend upon current and line constants only, but depend also upon the angle of phase displacement of the current delivered over the line. If = o, that is, non-inductive receiving circuit, FIG. 29. Locus of the generator and receiver e.m.fs. in a transmission line with varying load phase angle. E = - 4 EIz sin 21 ; that is, less than E + Iz, and thus the line drop is less than Iz. If 6 1, E Q is a maximum, = E + Iz, and the line drop is the impedance voltage. With decreasing 0, E decreases, and becomes = E; that is, no drop of voltage takes place in the line at a certain negative 60 ELEMENTS OF ELECTRICAL ENGINEERING value of which depends not only on z and 0i but on E and 7. Beyond this value of 6, E Q becomes smaller than E; that is, a rise of voltage takes place in the line, due to its reactance. This can be seen best graphically. Choosing the current vector 01 as the horizontal axis, for the same e.m.f. E received, but different phase angles 6, all vectors OE lie on a circle e with as center. Fig. 28. Vector OEz is constant for a given line and given current 7. Since E 3 E Q = OE = constant, E lies on a circle e Q with E s as center and OE = E as radius. To construct the diagram for angle 0, OE is drawn at the angle with 07, and EE Q parallel to ~OE*. The distance EE Q between the two circles on vector OEo is the drop of voltage (or rise of voltage) in the line. As seen in Fig. 29, E Q is maximum in the direction OE% as OE' , that is, for = , and is less for greater as well, OE" 'o, as smaller angles 6. It is = E in the direction 072"'o, in which case < 0, and minimum in the direction The values of E corresponding to the generator voltages E'Q, E" , E'" Q , # IV are shown by the points E' E" E f " E respectively. The voltages E" Q and E lv $ correspond to a wattless receiver cir- cuit E" and E. For non-inductive receiver circuit W the generator voltage is OE v o. 56. That is, in an inductive transmission line the drop of voltage is maximum and equal to Iz if the phase angle of the receiving circuit equals the phase angle of the line. The drop of voltage in the line decreases with increasing difference be- tween the phase angles of line and receiving circuit. It becomes zero if the phase angle of the receiving circuit reaches a certain negative value (leading current). In this case no drop of vol- tage takes place in the line. If the current in the receiving cir- cuit leads more than this value a rise of voltage takes place in the line. Thus by varying phase angle 9 of the receiving circuit the drop of voltage in a transmission line with current 7 can be made anything between Iz and a certain negative value. Or inversely the same drop of voltage can be produced for different values of the current 7 by varying the phase angle. / Thus, if means are provided to vary the phase angle of the receiving circuit, by producing lagging and leading currents at will (as can be done by synchronous motors or converters) , the voltage at the receiving circuit can be maintained constant IMPEDANCE OF TRANSMISSION LINES 61 within a certain range irrespective of the load and generator voltage. In Fig. 30 let OE = E, the receiving voltage; /, the power component of the line current; thus OE S = E s = Iz, the e.m.f. consumed by the power component of the current in the impe- dance. This e.m.f. consists of the e.m.f consumed by resistance ^Ei and the e.m.f. consumed by reactance OEz- FIG. 30. Regulation diagram for transmission line. Reactive components of the current are represented in the diagram in the direction OA when lagging and OB when leading. The e.m.f. consumed by these reactive components of the current in the impedance is thus in the direction e 1 '3, perpendicular to OEs- Combining OEz and OE gives the e.m.f. OE* which would be required for non-inductive load. If E Q is the generator voltage, E Q lies on a circle e Q with O^o as radius. Thus drawing E^E par- allel to e' z gives OE Q , the generator voltage; OE' S = EJZo, the 62 ELEMENTS OF ELECTRICAL ENGINEERING e.m.f. consumed in the impedance by the reactive component of the current; and as proportional thereto, OI' = I', the reactive current required to give at generator voltage E and power cur- rent 7 the receiver voltage E. This reactive current 7' lags be- hind E'z by less than 90 and more than zero degrees. 57. In calculating numerical values, we can pro'ceed either trigonometrically as in the preceding, or algebraically by resolv- ing all sine waves into two rectangular components; for instance, a horizontal and a vertical component, in the same way as in mechanics when combining forces. Let the horizontal components be counted positive toward the right, negative toward the left, and the vertical components positive upward, negative downward. Assuming the receiving voltage as zero line or positive hori- zontal line, the power current 7 is the horizontal, the wattless current I' the vertical component of the current. The e.m.f. con- sumed in resistance by the power current 7 is a horizontal com- ponent, and that consumed in resistance by the reactive current /' a vertical component, and the inverse is true of the e.m.f. consumed in reactance. We have thus, as seen from Fig. 30: Horizontal Vertical component component Receiver voltage, E, + E Power current, /, + 7 Reactive current, 7', HF 7' E.m.f. consumed in resistance r by the power current, Ir, + Ir E.m.f. consumed in resistance r by the reactive current, 7'r, + 7V E.m.f. consumed in reactance x by the power current, Ix, + Ix E.m.f. consumed in reactance x by the reactive current, I'x, I'x Thus, total e.m.f. required, or impressed e.m.f., Eo, E + Ir I'x + I'r + Ix; hence, combined, Eo = ^/(E + Ir or, expanded, + 2 E (Ir I'x) + (7 2 + 7' 2 )* 2 - IMPEDANCE OF TRANSMISSION LINES 63 From this equation I' can be calculated ; that is, the reactive current found which is required to give E and E at energy current 7. The lag of the total current in the receiver circuit behind the receiver voltage is tan = j. The lead of the generator voltage ahead of the receiver voltage is vertical component of EQ horizontal component of E Q B + Ir 7V and the lag of the total current behind the generator voltage is As seen, by resolving into rectangular components the phase angles are directly determined from these components. The resistance voltage is the same component as the current to which it refers. The reactance voltage is a component 90 time degrees ahead of the current. The same investigation as made here on long-distance trans- mission applies also to distribution lines, reactive coils, trans- formers, or any other apparatus containing resistance and reactance inserted in series into an alternating-current circuit. EXAMPLES 58. (1) An induction motor has 2000 volts impressed upon its terminals; the current and the power-factor, that is, the cosine of the angle of lag, are given as functions of the output in Fig. 31. The induction motor is supplied over a line of resistance r = 2.0 and reactance x = 4.0. (a) How must the generator voltage e Q be varied to maintain constant voltage e = 2000 at the motor terminals, and (b) At constant generator voltage CQ = 2300, how will the voltage at the motor terminals vary? 64 ELEMENTS OF ELECTRICAL ENGINEERING We have / a* ft . e = 2000. 63.4. = -i l(e + iz) 2 4 ezz sin 2 tan 0i = ~ = 2. cos = power-factor. Taking i from Fig. 31 and substituting, gives (a) the values of e for e = 2000, which are recorded in the table, and plotted in Fig. 31. JTPUT .10 20 30 40 50 60 70 80 90 100 110 .120 130 140 150 160 170 180 190 200 FIG. 31. Characteristics of induction motor and variation of generator e.m.f. necessary to maintain constant the e.m.f. impressed upon the motor. (6) At the terminal voltage of the motor e = 2000, the cur- rent is i, the output P, the generator voltage e Q . Thus at gen- erator voltage e'o = 2300, the terminal voltage of the motor is the current is and the power is 2300 . p , The values of e', i' , P' are recorded in the second part of the table under (6) and plotted in Fig. 32. IMPEDANCE OF TRANSMISSION LINES 65 (a) At e = 2000 Thus, M (6) Hence, at eo = 2300 Output, P = kw. Current, t Lag. Output, Current, ' Voltage, 12.0 84.3 2048 0. 13.45 2240 5 12.6 72.6 2055 6.25 14.05 2234 10 13.5 62.6 2060 12.4 15.00 2230 15 14.8 54.6 2065 18.6 16.4 2220 20 16.3 47.9 2071 24.4 18.0 2216 30 20.0 37.8 2084 36.3 22.0 2200 40 25.0 32.8 2093 48.0 27.5 2198 50 30.0 29.0 2110 59.5 32.7 2180 69 40.0 26.3 2146 78.5 42.8 2160 102 60.0 24.5 2216 110.2 62.6 2080 132 80.0 25.8 2294 131.0 79.5 1990 160 100.0 28.4 2382 149.0 96.4 1928 180 120.0 31.8 2476 156.5 111.5 1860 200 150.0 36.9 2618 155.0 132.0 1760 7 L IS 30 36 40 50 60 JO 80 90 100 110 120 130 !40 FIG. 32. Characteristics of induction motor, constant generator e.m.f. 5 66 ELEMENTS OF ELECTRICAL ENGINEERING 69. (2) Over a line of resistance r = 2.0 and reactance x = 6.0 power is supplied to a receiving circuit at a constant voltage of e = 2000. How must the voltage at the beginning of the line, or generator voltage, e Q , be varied if at no load the receiving circuit consumes a reactive current of i* 20 amp., this reac- tive current decreases with the increase of load, that is, of power current i\, becomes iz = at i\ = 50 amp., and then as leading current increases again at the same rate? REACT CURRE FIG. 33. Variation of generator e.m.f. necessary to maintain constant receiver voltage if the reactive component of receiver current varies propor- tional to the change of power component of the current. The reactive current, = 20 at = at and can be represented by = 0, = 50, * 2 = l - 20 = 20 -0.4 t i; the general equation of the transmission line is o = V (e H- iir -f i z x) = V(2000 + 2n + hence, substituting the value of z* 2 , (2*2 - e = V(2120 - 0.4 n) 2 + (40 - 6.8ti) a = V4,496,000 + 46.4 if - 2240 ^. ALTERNATING-CURRENT TRANSFORMER 67 Substituting successive numerical values for ii gives the values recorded in the following table and plotted in Fig. 33. ii eo '2120 20 2114 40 2116 60 2126 80 2148 100 2176 120 2213 140 2256 160 2308 180 2365 200 2430 13. ALTERNATING-CURRENT TRANSFORMER 60. The alternating-current transformer consists of one mag- netic circuit interlinked with two electric circuits, the primary circuit which receives energy, and the secondary circuit which delivers energy. Let TI = resistance, x\ = 2TrfSz = self-inductive or leakage reactance of secondary circuit, r = resistance, XQ = 2irfSi = self -inductive or leakage reactance of primary circuit, where S 2 and Si refer to that magnetic flux which is interlinked with the one but not with the other circuit. Let a ratio of - turns (ratio of transformation), primary An alternating e.m.f. E impressed upon the primary electric circuit causes a current, which produces a magnetic flux $ inter- linked with primary and secondary circuits. This flux gener- ates e.m.fs. EI and E { in secondary and in primary circuit, which Tjl are to each other as the ratio of turns, thus Ei = - Let E = secondary terminal voltage, I\ = secondary current, 0i = lag of current /i behind terminal voltage E (where B\ < denotes leading current). Denoting then in Fig. 34 by a vector OE = E the secondary 68 ELEMENTS OF ELECTRICAL ENGINEERING terminal voltage, 01 1 = l\ is the secondary current lagging by the angle EOI = 61. The e.m.f. consumed by the secondary resistance 7*1 is OE'i = E'i = Iiri in phase with /i. The e.m.f. consumed by the secondary reactance Xi is OE"\ = E'\ = I&i, 90 degrees ahead of /i. Thus the e.m.f. con- sumed by the secondary impedance z\ = Vn 2 + Xi 2 is the resultant of OE'i and OE"i, or OE"\ = E"\ =JiZi. OE'"\ combined with the terminal voltage OE = E gives the secondary e.m.f. OEi = E\. Proportional thereto by the ratio of turns and in phase there- FIG. 34. Vector diagram of e.m.fs. and currents in a transformer. with is the e.m.f. generated in the primary OEi = E f where To generate e.m.f. EI and E i} the magnetic flux 0$ = is required, 90 time degrees ahead of OE\ and OEi. To produce flux $ the m.m.f. of F ampere-turns is required, as determined from the dimensions of the magnetic circuit, and thus the primary current /oo, represented by vector O/oo, leading 0$ by the angle a. Since the total m.m.f. of the transformer is given by the primary exciting current 7 o, there must be a component of primary current /', corresponding to the secondary current /i, which may be called the primary load current, and which is ALTERNATING-CURRENT TRANSFORMER 69 opposite thereto and of the same m.m.f.; that is, of the intensity /' = a/i, thus represented by vector 01' = I' = a/i. O/oo, the primary exciting current, and the primary load current O/', or component of primary current corresponding to the secondary current, combined, give the total primary current O/o = /o- The e.m.f. consumed by resistance in the primary is OE' Q = E' Q = / O r in phase with / . The e.m.f. consumed by the primary reactance is OE"o = E"Q = / o, 90 degrees ahead of O/o. OE'Q and OE"o combined gives OE'" Q , the e.m.f. consumed by the primary impedance. FIG. 35. Vector diagram of transformer with lagging load current. Equal and opposite to the primary counter-generated e.m.f. OEi is the component of primary e.m.f., OE f , consumed thereby. OE' combined with OE"' Q gives OE Q = E Qj the primary im- pressed e.m.f., and angle 0o = -#o#/o, the phase angle of the primary circuit. Figs. 35, 36, and 37 give the polar diagrams of 0i = 45 or lagging current, 0i = zero or non-inductive circuit, and 6 = 45 or leading current. 61. As seen, the primary impressed e.m.f. E required to pro- duce the same secondary terminal voltage E at the same current 1 1 is larger with lagging or inductive and smaller with leading 70 ELEMENTS OF ELECTRICAL ENGINEERING current than on a non-inductive secondary circuit; or, inversely, at the same secondary current I\ the secondary terminal voltage E with lagging current is less and with leading current more than with non-inductive secondary circuit, at the same primary impressed e.m.f. EQ. The calculation of numerical values is not practicable by measurement from the diagram, since the magnitudes of the different quantities are too different, E'i:E"iiEi:Eo being frequently in the proportion 1 : 10 : 100 :2000. Trigonometrically, the calculation is thus: FIG. 36. Vector diagram of transformer with non-inductive loading. In triangle OEEi, Fig. 34, writing tan 0' = ^ we have, also, OEi 2 = OE 2 + EEi 2 - 2 OE EEi cos EEi = i = 180 - tf + 0i, hence, E = E 2 + 7i 2 zi 2 + 2 #7iZi cos (0' - 00 . This gives the secondary e.m.f., EI, and therefrom the primary counter-generated e.m.f. Ei = In triangle EOEi we have sin E\OE -T- sin E\EO = .E^n -f- EiO ALTERNATING-CURRENT TRANSFORMER 71 thus, writing E&E = 0", we have sin 0" -4- sin (0' - 0i) = hz + Ei, wherefrom we get % 6", and E 1 OI l = 6 = 0, + 0", the phase displacement between secondary current and secondary e.m.f. FIG. 37. Vector diagram of transformer with leading load current. In triangle O/oo/o we have since and 0/ 2 = O/oo 2 + /oo/o 2 - 2 O/oo/oo/o COS O/oo/o, #i00 = 90, $ O/oo/o = 90 + + a, J 00 / = r = al O/oo = IQQ = exciting current, calculated from the dimensions of the magnetic circuit. Thus the primary current is J 2 = J 00 2 + a 2/ l2 _{_ 2 a/ i/oo sin (0 + a). In triangle O/oo/o we have sin /ooO/o -T- sin O/oo/o = /oo/o -^ 0/ ; writing this becomes sin 0" -^ sin (0 + a) = a/i -?- / ; therefrom we get 0" , and thus #'0/ = 2 = 90 - a - 0" . 72 ELEMENTS OF ELECTRICAL ENGINEERING In triangle OE'E Q we have OE Q 2 = OE~' 2 + WWo* -20E' WW cos OE' # ; writing tan 0' = ' 7*0 we have OE'E = 180 - 0' + 2 , OE' = E< = > thus the impressed e.m.f. is ET 9 ^l 2 I T 9 9 I *^UO*0 / / s V = ^2" + ^0 2 ^0 2 H COS (0 2 ). In triangle OE'E Q sin E'OEo -T- sin thus, writing we have sin Q'\ + sin (0' - 2 ) = /oo ^ ^ ; herefrom we get ^ 0"i, and ^ 00 = 02 + 0"l, the phase displacement between primary current and impressed e.m.f. As seen, the trigonometric method of transformer calculation is rather complicated. 62. Somewhat simpler is the algebraic method of resolving into rectangular components. Considering first the secondary circuit, of current 7i lagging behind the terminal voltage E by angle 0i. The terminal voltage E has the components E cos 0i in phase, E sin 0i in quadrature with and ahead of the current /i. The e.m.f. consumed by resistance r 1} I-pi, is in phase. The e.m.f. consumed by reactance x\, I&i, is in quadrature ahead of /i. Thus the secondary e.m.f. has the components E cos 0i + /ifi in phase, E sin 0i + IiXi in quadrature ahead of the current /i, and the total value, Ei = V(E cos 0i + /in) 2 + (E sin 0i + /iZi) 2 , ALTERNATING-CURRENT TRANSFORMER 73 and the tangent of the phase angle of the secondary circuit is E sin 0i + tan e = E COS 61 + Resolving all quantities into components in phase and in quadrature with the secondary e.m.f. EI, or in horizontal and in vertical components, choosing the magnetism or mutual flux as vertical axis, and denoting the direction to the right and upward as positive, to the left and downward as negative, we have Horizontal Vertical component component Secondary current, /i, Ii cos + /i sin Secondary e.m.f., EI, EI Primary counter-generated e.m.f., E l = ^ - - 1 a a Primary e.m.f. consumed thereby, E' = - E,, + ^ Primary load current, /' = a/i,+ al\ cos 6 al\ sin Magnetic flux, $>, < Primary exciting current, / O o, con- sisting of core loss current, /oo sin a magnetizing current, /oo cos a. hence, total primary current, J , Horizontal component Vertical component all cos 0i + /oo sin a (all sin 0i + /oo cos a) E.m.f. consumed by primary resistance r , E'Q = Ior in phase with /o, Horizontal component Vertical component r a/i cos + r /oo sin a (r a/i sin + r /oo cos a) E.m.f. consumed by primary reactance x , E = IoX Q , 90 ahead of /o, Horizontal component Vertical component X ali sin + Zo/oo cos a + X ali cos + rr /oo sin a Tjl E.m.f. consumed by primary generated e.m.f., E' = - horizontal. 74 ELEMENTS OF ELECTRICAL ENGINEERING The total primary impressed e.m.f., E Q , Horizontal component ET + ol\ (r cos + #o sin 6) + /oo (TO sin a + X Q cos a). Vertical component a/i (r sin cos 0) + 7 o (ro cos a XQ sin a), or writing tan 0'o = > ?*o since \A*o 2 + 2 = ZQ, sin 0' = , and cos 0' = -' 20 ZQ Substituting this value, the horizontal component of E is pi + a2 /i cos (0 0' ) + 2o/oo sin (a + r ) ; the vertical component of E is azo/i sin (0 0' ) + 2 /oo cos (a + 0' ), and, the total primary impressed e.m.f. is o=\/r +azo/icos(0 0'o)+*oloosin(a+0'o)] > + 2azo/oo . . , o*zo 2 /i 2 , a 2 2o 2 /oo 2 . 2a"zo 2 /i/oo . Mn(a+g / o)+ - + + - Combining the two components, the total primary current is + /oo sin a) 2 + (a/i sin + /oo cos a) Since the tangent of the phase angle is the ratio of vertical component to horizontal component, we have, primary e.m.f. phase, _ azp/i sin (0 0' ) -f ZO/OQ cos ( + 0'o) tan -j=, + 020/1 cos (0 - 0' ) + 2 /oo sin (a - 0' ) primary current phase, and lag of primary current behind impressed e.m.f., _ ali sin -f /oo cos a ~ ali cos + /oo sin a ALTERNATING-CURRENT TRANSFORMER 75 EXAMPLES 63. (1) In a 20-kw. transformer the ratio of turns is 20 -f- 1, and 100 volts is produced at the secondary terminals at full load. What is the primary current at full load, and the regu- lation, that is, the rise of secondary voltage from full load to no load, at constant primary voltage, and what is this primary voltage? (a) at non-inductive secondary load, (6) with 60 degrees time lag in the external secondary circuit, (c) with 60 degrees time lead in the external secondary circuit. The exciting current is 0.5 amp., the core loss 600 watts, the primary resistance 2 ohms, the primary reactance 5 ohms, the secondary resistance 0.004 ohm, the secondary reactance 0.01 ohm. Exciting current and core loss may be assumed as constant. 600 watts at 2000 volts gives 0.3 amp. core loss current, hence V0.5 2 3 2 = 0.4 amp. magnetizing current. We have thus r = 2 XQ 5 = 0.004 = 0.01 /oo cos a = 0.3 /oo sin a = 0.4 /oo = 0.5 0.05 1. Secondary current as horizontal axis: Non-inductive, 0i =0 01 -*+W Le*d, l = - 60 Hor. Vert. Hor. Vert. Hor. Vert. Secondary current, /i. . Secondary terminal voltage, E. 200 100 0.8 100.8 % +2.0 +2.0 200 50 0.8 50.8 +86.6 + 2.0 +88.6 200 50 0.8 50.8 -86.6 + 2.0 -84.6 Resistance voltage, I\T\. Reactance voltage, I\xi. Secondary e.m.f ., E\... Secondary e.m.f., total tan 100.80 +0.0198 + 1.1 102.13 + 1.745 +60.2 98.68 - 1.665 -59.0 76 ELEMENTS OF ELECTRICAL ENGINEERING 2. Magnetic flux as vertical axis: Non-inductive, 61 = Lag, 0! = + 60 Lead, 0i =- 60 Hor. Vert. Hor. Vert. Hor. Vert. Secondary gen- erated e.m.f., E -100.80 -200 + 10 0.3 + 10.3 20.6 3.0 2016 2039.6 + 4 + 0.2 0.4 - 0.6 1.2 +51.3 +50.1 -102.13 - 99.4 + 4.97 0.3 + 5.27 10.54 45.20 2042.6 2098.34 -172.8 - 8.64 - 0.4 - 9.04 - 18.08 + 26.35 + 8.27 - 98.68 -103 + 5.15 0.3 + 5.45 10.90 - 40.85 1973.6 1943.65 -171.4 + 8.57 - 0.4 + 8.17 + 16.34 +27.25 +43.59 Secondary cur- rent, 1 1 Primary load current, /' = -a/i Primary excit- ing current, 7 o Total primary current, /o . . . . Primary resist- ance, voltage, /oPo Primary react- ance, voltage, Iox E.m.f. consum- ed by primary counter e.m.f., -E l a Total primary impressed e.m.f., E Hence, Non-inductive, 0i = Lag, 0i = + 60 Lead, 0! = - 60 Resultant Eo .... 2040 1 2098 3 1944 2 Resultant /o 10 32 10 47 9 82 Phase of E -1.4 - 2 1 2 Phase of /o +3.3 +59 8 56 3 Primary lag 60 +4.7 +60 55 1 W Rpsrulation 1 02005 1 04915 9721 2000' ' Drop of voltage, per cent 2 005 4 915 2 79 Change of phase, do 0i 4.7 4.9 RECTANGULAR COORDINATES 77 14. RECTANGULAR COORDINATES 64. The vector diagram of sine waves gives the best insight into the mutual relations of alternating currents and e.m.fs. For numerical calculation from the vector diagram either the trigonometric method or the method of rectangular components is used. The method of rectangular components, as explained in the above paragraphs, is usually simpler and more convenient than the trigonometric method. In the method of rectangular components it is desirable to distinguish the two components from each other and from the resultant or total value by their notation. To distinguish the components from the resultant, small letters are used for the components, capitals for the resultant. Thus in the transformer diagram of Section 13 the secondary current I\ has the horizontal component ii = I\ cos 0i, and the vertical component i'\ + I\ sin 0\. To distinguish horizontal and vertical components from each other, either different types of letters can be used, or indices, or a prefix or coefficient. Different types of letters are inconvenient, indices distinguish- ing the components undesirable, since indices are reserved for distinguishing different e.m.fs., currents, etc., from each other. Thus the most convenient way is the addition of a prefix or coefficient to one of the components, and as such the letter j is commonly used with the vertical component. Thus the secondary current in the transformer diagram, Section 13, can be written i\ + ji* = Ii cos 0i + jli sin 0i. (1) This method offers the further advantage that the two com- ponents can be written side by side, with the plus sign between them, since the addition of the prefix j distinguishes the value jit or jli sin 0i as vertical component from the horizontal com- ponent i\ or 1 1 cos 0i. 1 1 = ii + ji* (2) thus means that I\ consists of a horizontal component i\ and a vertical component iz, and the plus sign signifies that i\ and iz are combined by the parallelogram of sine waves. 78 ELEMENTS OF ELECTRICAL ENGINEERING The secondary e.m.f. of the transformer in Section 13, Fig. 34, is written in this manner, E\ = ei, that is, it has the hori- zontal component e\ and no vertical component. The primary generated e.m.f. is E '==' 00 and the e.m.f. consumed thereby E ' = + e i- w The secondary current is where ii = Ii cos 0i, iz = Ii sin 0i, (6) and the primary load current corresponding thereto is I' = - aii = aii - jaiz. (7) The primary exciting current, Joo = h - jg, (8) where h = J o sin a is the hysteresis current, g = I o cos a the reactive magnetizing current. Thus the total primary current is J = I' + J 00 = (aii + h) -j (aiz + g). (9) The e.m.f. consumed by primary resistance r Q is r Jo = TQ (aii + h) - jr (aiz + 0). (10) The horizontal component of primary current (aii + h) gives as e.m.f. consumed by reactance XQ a negative vertical com- ponent, denoted by JXQ (aii + h). The vertical component of primary current j (aiz + g) gives as e.m.f. consumed by react- ance XQ a positive horizontal component, denoted by XQ (aiz + (/) Thus the total e.m.f. consumed by primary reactance XQ is XQ (aiz + g) + jx Q (aii + h), (11) and the total e.m.f. consumed by primary impedance is r (aii + A) + x (aiz + g) - j[r Q (aiz + g) - XQ (aii + h)]. (12) RECTANGULAR COORDINATES 79 Thus, to get from the current the e.m.f. consumed in react- ance X Q by the horizontal component of current, the coefficient j has to be added; in the vertical component the coefficient j omitted; or, we can say the reactance is denoted by jx Q for the horizontal and by r- for the vertical component of current. In other words, if 7 = i ji' is a current, x the reactance of its circuit, the e.m.f. consumed by the reactance is jxi H- xi' = xi' + jxi. 65. If instead of omitting j in deriving the reactance e.m.f. for the vertical component of current we would add j also (as done when deriving the reactance e.m f. for the horizontal component of current), we get the reactance e.m.f. jxi fxi', which gives the correct value jxi + xi', if f = - 1; (13) that is, we can say, in deriving the e.m.f. consumed by reactance, x, from the current, we multiply the current by jx, and substitute By defining, and substituting, j 2 = 1, jx can thus be called the reactance in the representation in rectangular coordinates and r -+- jx the impedance. The primary impedance voltage of the transformer in the preceding could thus be derived directly by multiplying the current, /o = (aii + h) - j (aii + g), (9) by the impedance, Z = r -f jx Q , which gives E'o = Zo/o = (r + jx<>) [(aii + h) - j (ai 2 + g)] = r (aii + h) - jr Q (ai 2 + g) + jx Q (aii + h) - j 2 x Q (ai 2 -f g), and substituting j 2 = 1, E'o = [r (aii + h) + XQ (ai 2 + g)] - j [r (ai 2 + g) - X Q (aii + h)], (14) 80 ELEMENTS OF ELECTRICAL ENGINEERING and the total primary impressed e.m.f. is thus EQ = E -[" E o = [^ + r (ail + h) + x ( + i' 2 > (18) The capital letter I in the symbolic expression / = i + ji f thus represents more than the / used in the preceding for total current, etc., and gives not only the intensity but also the phase. It is thus necessary to distinguish by the type of the latter the capital letters denoting the resultant current in symbolic expres- sion (that is, giving intensity and phase) from the capital letters giving merely the intensity regardless of phase; that is, I = denotes a current of intensity / = and phase tan = . ^ RECTANGULAR COORDINATES 81 In the following, dotted italics wfll be used for the symbolic expressions and plain italics for the absolute values of alternating waves. In the same way z = \/r 2 + x 2 is denoted in symbolic repre- sentation of its rectangular components by Z = r + jx. (91) When using the symbolic expression of rectangular coordinates it is necessary ultimately to reduce to common expressions. ; Thus in the above discussed transformer the symbolic expres- sion of primary impressed e.m.f. E Q = |j^ + r Q (aii + h) + X (ai 2 +g) J -j [r (ai 2 +0) -z (a*'i+/i)J (15) means that the primary impressed e.m.f. has the intensity (ai' 2 +flf)J (20) and the phase tan 0o = - 1 + r (aii + h) + X (ai z + flf) This symbolism of rectangular components is the quickest and simplest method of dealing with alternating-current phenom- ena, and is in many more complicated cases the only method which can solve the problem at all, and therefore the reader must become fully familiar with this method. EXAMPLES 67. (1) In a 20-kw. transformer the ratio of turns is 20 : 1, and 100 volts are required at the secondary terminals at full load. What is the primary current, the primary impressed e.m.f., and the primary lag, (a) at non-inductive load, 0i = 0; (6) with 0i = 60 degrees time lag in the external secondary circuit; (c) with 61 = 60 degrees time lead in the external secondary circuit? 82 ELEMENTS OF ELECTRICAL ENGINEERING i i !s 32 i t* s + + I II OS 2 o * d 111 + 3" O O (NO do CO CO ^ 1 fe : + : ^J- f 11 + ^ a l' ^ E 8 * & S ill!! i D a c S D. g .S : s z 1 & So. : : S t> ' : tf 05 n s s 'N o. -S V 3 3 & ** 03 T3 S fl 'E i ja l?l> +* + +N S5S -! oo 1 d S| H " 84 ELEMENTS OF ELECTRICAL ENGINEERING The exciting current is /' 00 = 0.3 0.4 j amp. at e = 2000 volts impressed, or rather, primary counter-generated e.m.f. The primary impedance, Z = 2 -f 5 j ohms. The secondary impedance, Z\ = 0.004 + 0.01 j ohm. We have, in symbolic expression, choosing the secondary current /i as real axis, the results calculated in tabulated form on page 82. 68. (2) e Q = 2000 volts are impressed upon the primary circuit of a transformer of ratio of turns 20:1. The primary impedance is Z Q = 2 -f 5 j, the secondary impedance, Zi = 0.004 + 0.01 j, and the exciting current at e r = 2000 volts counter-generated e.m.f. is 7 o = 0.3 0.4 j; thus the exciting admittance, Y = ^ = (0.15 - 0.2 j)10~ 3 . 6 What is the secondary current and secondary terminal voltage and the primary current if the total impedance of the secondary circuit (internal impedance plus external load) consists of (a) resistance, Z = r = 0.5 non-inductive circuit. (6) impedance, Z = r + jx = 0.3 + 0.4 j inductive circuit. (c) impedance, Z = r -\- jx = 0.3 0.4 j anti-inductive circuit. Let e = secondary e.m.f., assumed as real axis in symbolic expression, and carrying out the calculation in tabulated form, on page 83. 69. (3) A transmission line of impedance Z = r -}- jx = 20 + 50 j ohms feeds a receiving circuit. At the receiving end an apparatus is connected which produces reactive lagging or leading currents at will (synchronous machine) ; 12,000 volts are impressed upon the line. How much lagging and leading currents respectively must be produced at the receiving end of the line to get 10,000 volts (a) at no load, (6) at 50 amp. power current as load, (c) at 100 amp. power current as load? Let e = 10,000 = e.m.f. received at end of line, ii = power current, and i% = reactive lagging current; then total line current. LOAD CHARACTERISTIC OF TRANSMISSION LINE 85 The voltage at the generator end of the line is then E = e + ZI = e + (r + jx) (ii ji z ) = (e + rii + xi 2 ) j (n' 2 xii) = (10,000 + 20 ti + 50 1*2) - j (20 z a - 50t'i); or, reduced, f-ni + zi 2 ) 2 + (ri2 - thus, since E = 12,000, 12,000 = V(10,000 + 20*i + 50^) 2 + (20 i z - 50^i) 2 . . (a) At no load i\ = 0, and 12,000 = hence, i 2 = -j- 39.5 amp., reactive lagging current, I = 39.5 ./. (6) At half load ii = 50, and 12,000 = VuMXX) + 50i 2 ) 2 + (20z 2 - 2500) 2 ;- hence, is = + 16 amp., lagging current,/ = 50 16 j. (c) At full load ii = 100, and 12,000 = V(12,000 + 50i 2 ) 2 + (20 i - 5000) 2 ; hence, i 2 = - 27.13 amp., leading current, I = 100 + 27.13 j. 15. LOAD CHARACTERISTIC OF TRANSMISSION LINE 70. The load characteristic of a transmission line is the curve of volts and watts at the receiving end of the line as function of the amperes, and at constant e.m.f . impressed upon the generator end of the line. Let r = resistance, x = reactance of the line. Its impedance z = -y/r 2 + x 2 can be denoted symbolically by Z = r + jx. Let EQ = e.m.f. impressed upon the line. Choosing the e.m.f. at the end of the line as horizontal com- ponent in the vector diagram, it can be denoted by E = e. 86 ELEMENTS OF ELECTRICAL ENGINEERING At non-inductive load the line current is in phase with the e.m.f. e, thus denoted by 7 = i. The e.m.f. consumed by the line impedance Z r + jx is E! = ZI = (r + jx) i = ri+jxi. (1) Thus the impressed voltage, ' Eo = E + Ei = e + ri + ja. (2) or, reduced, #o = V(e + n) 2 + z 2 * 2 , (3) and _ 6 = ^o 2 - z 2 * 2 - n, the e.m.f. (4) p = d = i V-Eo 2 - x 2 i 2 - ri 2 , (5) the power received at end of the line. The curve of e.m.f. e is an arc of an ellipse. With open circuit i = 0, e = E and P = 0, as is to be expected. At short circuit, e = 0, = \/#o 2 x z i 2 ri, and ; (6) X' that is, the maximum line current which can be established with a non-inductive receiver circuit and negligible line capacity. 71. The condition of maximum 'power delivered over the line ' i| f-* on that is, substituting (3): '! V#o 2 - x*i* = e + ri, and expanding, gives e* = (r 2 + x 2 ) i 2 (8) = z 2 i 2 ; hence, e zi, and - = z. (9) -T- = 7*1 is the resistance or effective resistance of the receiving circuit; that is, the maximum power is delivered into a non- LOAD CHARACTERISTIC OF TRANSMISSION LINE 87 inductive receiving circuit over an inductive line upon which is impressed a constant e.m.f., if the resistance of the receiving circuit equals the impedance of the line, TI = z. In this case the total impedance of the system is Z = Z + n = r + z + jx, (10) or, zo = V(r + z) 2 + z 2 . (11) Thus the current is *o V(r + z) 2 + x 2 and the power transmitted is Eo 2 z (r that is, the maximum power which can be transmitted over a line of resistance r and reactance x is the square of the impressed e.m.f. divided by twice the sum of resistance and impedance of the line. At x = 0, this gives the common formula, Inductive Load 72. With an inductive receiving circuit of lag angle 6, or power-factor p = cos 8, and inductance factor q = sin 6, at e.m.f. E = e at receiving circuit, the current is denoted by I = I(p-jq); (15) thus the e.m.f. consumed by the line impedance Z = r -f jx is E! = ZI = I (p -jq)(r+jx) = I [(rp + xq) - j (rq - xp)], and the generator voltage is Eo = E + #1 = [e + / (rp + sg)]. - jl (rq - xp); (16) 88 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, #o = V+ 7 (rp + xq)} 2 + P (rq - xp)*, (17) and e = \/E Q *- P(rq-xp) 2 - I (rp + xq). (18) The power received is the e.m.f. times the power component of the current; thus P = elp __ = Ip \/Eo*- P(rq-xp)* - Pp (rp + xq). (19) The curve of e.m.f., e, as function of the current I is again an arc of an ellipse. At short circuit e = 0; thus, substituted, /- (20) the same value as with non-inductive load, as is obvious. 73. The condition of maximum output delivered over the line is that is, differentiated, V#o 2 -I 2 (rq-xp) 2 = e + I (rp + xq); (22) substituting and expanding, e = Iz] or y = z. (23) Zi = -j is the impedance of the receiving circuit; that is, the power received in an inductive circuit over an inductive line is a maximum if the impedance of the receiving circuit, z\ y equals the impedance of the line, z. In this case the impedance of the receiving circuit is Zi = z(p +jq), (24) and the total impendance of the system is ZQ = Z -{- Zi = r + jx + z (p + jq) LOAD CHARACTERISTIC OF TRANSMISSION LINE 89 Thus, the current is /i = and the power is V(r-f 2 (z + rp + xq) EXAMPLES (25) (26) 74. (1) 12,000 volts are impressed upon a transmission line of impedance Z = r + jx = 20 + 50 j. How do the voltage \ \ \ V son mo VOLTS 11000 9000 7000 4000 20 40 60 80 100 120 140 160 .180 200 220 FIG. 39. Non-reactive load characteristic^ of a transmission line. Con- stant impressed e.m.f. and the output in the receiving circuit vary with the current with non-inductive load? Let e = voltage at the receiving end of the line, i = current: thus = ei power received. The voltage impressed upon the line is then Eo = e + Zi = e +ri + jxi; 90 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, Eo = V( Since E Q = 12,000, 12,000 = V(e + n) 2 + xH* = V(e + 20 i)* + 2500 i\ e = V12,000 2 - x 2 i 2 - ri = Vl2,000 2 - 2500^ - 20 The maximum current for e = is thus, = V12,000 2 - 2500 i 2 - 20 i\ i = 223. Substituting for i gives the values plotted in Fig. 39. i e p = ei 12,000 20 11,500 230 X 10 3 40 11,000 440 X 10 3 60 10,400 624 X 10 3 80 9,700 776 X 10 3 100 8,900 890 X 10 3 120 8,000 960 X 10 3 140 6,940 971 X 10 3 160 5,750 920 X 10 3 180 4,340 784 X 10 3 200 2,630 526 X 10 3 220 400 88 X 10 3 223 16. PHASE CONTROL OF TRANSMISSION LINES 76. If in the receiving circuit of an inductive transmission line the phase relation can be changed, the drop of voltage in the line can be maintained constant at varying loads or even decreased with increasing load; that is, at constant generator voltage the transmission can be compounded for constant voltage at the receiving end, or even over-compounded for a voltage increasing with the load. 1. Compounding of Transmission Lines for Constant Voltage Let r = resistance, x = reactance of the transmission line, CQ = voltage impressed upon the beginning of the line, e = vol- tage received at the end of end line. PHASE CONTROL OF TRANSMISSION LINES 91 Let i = power current in the receiving circuit; that is, P ei = transmitted power, and ii = reactive current produced in the system for controlling the voltage. i\ shall be considered positive as lagging, negative as leading current. Then the total current, in symbolic representation, is / = i - jii; the line impedance is Z = r + jx, and thus the e.m.f. consumed by the line impedance is Ei = ZI = (r + jx) (i - jii) = ri + jrii + jxi - J 2 xii; and substituting f 1, Ei = (ri + xii) - j (rii - xi). Hence the voltage impressed upon the line Eo = e 4- Ei = (e + ri + xii) - j (rii - xi) ; (1) or, reduced, _ eo = V(e + ri + xii)* + (n\ - xi)*. (2) If in this equation e and e Q are constant, ii, the reactive com- ponent of the current, is given as a function of the power com- ponent current i and thus of the load ei. Hence either e Q and e can be chosen, or one of the e.m.fs. e Q or e and the reactive current ii corresponding to a given power current i. 76. If ii = with i = 0, and e is assumed as given, e Q = e. Thus, _ e = V(e + ri + xi^ + (rii - xi)*; 2 e (ri + xii) + (r 2 + x 2 ) (i 2 + ii 2 ) = 0. From this equation it follows that ex Ve 2 x 2 - 2 eriz 2 - i 2 z* /ox *i - - - i~ Thus, the reactive current ii must be varied by this equation to maintain constant voltage e = eo irrespective of the load ei. As seen, in this equation, ii must always be negative, that is, the current leading. 92 ELEMENTS OF ELECTRICAL ENGINEERING ii becomes impossible if the term under the square root becomes negative, that is, at the value e 2 x 2 - 2 eriz 2 - i 2 z 4 = 0; 1 ,' f-^4 - (4) At this point the power transmitted is This is the maximum power which can be transmitted with- Q (g _ Y\ out drop of voltage in the line, with a power current i = - ^ . The reactive current corresponding hereto, since the square root becomes zero, is III ,-.....'.; ti = 5[; . . . (6) thus the ratio of reactive to power current, or the tangent of the phase angle of the receiving circuit, is A larger amount of power is transmitted if e is chosen > e, a smaller amount of power if e Q < e. In the latter case ii is always leading; in the former case i\ is lagging at no load, becomes zero at some intermediate load, and leading at higher load. 77. If the line impedance Z r + fa and the received voltage e is given, and the power current ^o at which the reactive current shall be zero, the voltage at the generator end of the line is determined hereby from the equation (2) : e Q = V(e -f ri + xii) 2 + (ri { - xi) 2 , by substituting i\ = 0, i = z'o, Substituting this value in the general equation (2) : e = V(e + ri gives (e + n ) 2 + zV = (e + ri + xitf + (rii - xi) 2 (9) as equation between i and i\. PHASE CONTROL OF TRANSMISSION LINES 93 If at constant generator voltage e : at no load, i = 0, e = e , i\ = i'o, and at the load, (10) i = i o, 6 = BO, i\ = it is, substituted: no load, load io, Thus, (eo + fc'o) + #V| = (eo + n'o) 2 + x 2 t*o 2 ; or, expanded, iV(r 2 + x 2 ) + 2 i' xe = io 2 (r 2 + x 2 ) + 2 i Q re . (13) This equation gives i'o as function of io, e , r, x. If now the reactive current i\ varies as linear function of the power current i, as in case of compounding by rotary converter with shunt and series field, it is Substituting this value in the general equation (eo + n ) 2 + *V = (e + ri + ai) + (rii - xz) 2 gives e as function of i; that is, gives the voltage at the receiving end as function of the load, at constant voltage 60 at the gener- ating end, and e = eo for no load, i = 0, ii = i'o, and e = e Q for the load, i = io, ii = 0. Between i = and i = io, e > eo, and the current is lagging. Above i = io, e < e Q , and the current is leading. By the reaction of the variation of e from eo upon the receiving apparatus producing reactive current z'i, and by magnetic satura- tion in the receiving apparatus, the deviation of e from eo is reduced, that is, the regulation improved. 2. Over-compounding of Transmission Lines 78. The impressed voltage at the generator end of the line was found in the preceding, eo = V(e 4- ri + ai) a + (rii - xi) 2 . (2) 94 ELEMENTS OF ELECTRICAL ENGINEERING If the voltage at the end of the line e shall rise proportionally to the power current i, then e = e\ + ai', (15) thus, eo = V[ej. + (a + r) t + ai] + (n'i - **) 2 , and herefrom in the same way as in the preceding we get the characteristic curve of the transmission. If eo e\ t i\ = at no load, and is leading at load. If o < ei, ii is always leading, the maximum output is less than before. If eo > ei, i\ is lagging at no load, becomes zero at some inter- mediate load, and leading at higher load. The maximum output is greater than at e = e\. The greater a, the less is the maximum output at the same GO and \. The greater eo, the greater is the maximum output at the same e\ and a, but the greater at the same time the lagging current (or less the leading current) at no load. EXAMPLES 79. (1) A constant voltage of e is impressed upon a trans- mission line of impedance Z = r + jx = 10 + 20 j. The vol- tage at the receiving end shall be 10,000 at no load as well as at full load of 75 amp. power current. The reactive current in the receiving circuit is raised proportionally to the load, so as to be lagging at no load, zero at full load or 75 amp., and lead- ing beyond this. What voltage e has to be impressed upon the line, and what is the voltage e at the receiving end at J, %, and 1J load? Let J = ii ji z = current, E = e voltage in receiving circuit. The generator voltage is then Eo = e + ZI = e + (r + jx) (ii - ji 2 ) = (e + rii + xi z ) j (n' 2 xii) = (e + 10 n + 20 i 2 ) - j (10 t, - 20 t'O; or, reduced, e Q * = ( e + n'j + x i z y 4. ( n ' 2 _ ^2. = (e + 10 ii + 20 tj) J + (10 t, - 20 *i) 2 . When t'i = 75, t, = 0, e = 10,000; PHASE CONTROL OF TRANSMISSION LINES 95 substituting these values, e 2 = 10,750 2 + 1500 2 = 117.81 X 10 6 ; eo = 10,860 volts is the generator voltage. hence, When ii = 0, e = 10,000, e Q = 10,860, let i 2 = i: these values substituted give 117.81 X 10 6 = (10,000 + 20 i) 2 + 100 i 2 = 100 X 10 6 + 400 i X 10 3 + 500 * 2 , or, i = 44.525 - 1.25 i 2 10~ 3 ; this equation is best solved by approximation, and then gives p = 42.3 amp. reactive lagging current at no load. Since eo 2 = (e + rii + xi 2 ) 2 + (ri z - xii) 2 , it follows that e = Veo 2 (*2 xii) 2 (rii + 2^2); or, e = V117.81 + 10 6 - (10 i* - 20 n) 2 - (10 ii + 20 t 2 ). Substituting herein the values of ii and i 2 gives e. t'l tz e 42.3 10,000 25 28.2 10,038 50 14.1 10,038 75 10,000 100 -14.1 9,922 125 -28.2 9,803 80. (2) A constant voltage e Q is impressed upon a trans- mission line of impedance Z = r + jx = 10 + 10 j. The vol- tage at the receiving end shall be 10,000 at no load as well as at full load of 100 amp. power current. At full load the total current shall be in phase with the e.m.f. at the receiving end, and at no Load a lagging current of 50 amp. is permitted. How much additional reactance x is to be inserted, what must be the generator voltage e , and what will be the voltage e at the receiv- 96 ELEMENTS OF ELECTRICAL ENGINEERING ing end at % load and at 1J^ load, if the reactive current varies proportionally with the load? Let XQ = additional reactance inserted in circuit. Let I i\ jiz = current. Then e 2 = (e + rii + x^) 2 + (n a - Xiii) 2 = (e + 10 ii + x^Y + (10 i 2 - where Xi = x + #o = total reactance of circuit between e and e Q . At no load, ii = 0, i 2 = 50, e = 10,000; thus, substituting, e 2 = (10,000 + 50 zi) 2 + 250,000. At full load, ii = 100, i a = 0, e = 10,000; thus, substituting, eo 2 = 121 X 10 6 + 10,000 xj. Combining these gives (10,000 + 50 zi) 2 + 250,000 = 121 X 10 6 + 10,000 a^i 2 ; hence, xi = 66.5 + 40.8 = 107.3 or 25.7; thus X Q = Xi x = 97.3 or 15.7 ohms additional reactance. Substituting xi = 25.7 gives e 2 = ( e + 10 ^ + 25.7 i 2 ) 2 + (10 i* - 25.7 ii) 2 , but at full load ii = 100, i 2 = 0, e = 10,000, which values substituted give e 2 = 121 X 10 6 + 6.605 X 10 6 = 127.605 X 10 6 , eo = 11,300, generator voltage. Since e = vV - (10 12 - 25.7 *i) 2 - (10 ii + 25.7 i 2 ), it follows that 6 = V127.605 X 10 6 - (10 i t - 25.7 ii) 2 - (10 ii + 25.7 t 2 ). Substituting for ii and z* 2 gives e. PHASE CONTROL OF TRANSMISSION LINES 97 u ti e 50 10,000 50 25 10,105 100 10,XKX) 150 -25 9,658 81. (3) In a circuit whose voltage e fluctuates by 20 per cent, between 1800 and 2200 volts, a synchronous motor of internal impedance Z = r + jx = 0.5 + 5 j is connected through a reactive coil of impedance Z\ = r\ + jx\ = 0.5 -f- 10 j and run light, as compensator (that is, generator of reactive currents). How will the voltage at the synchronous motor terminals e\, at constant excitation, that is, constant counter e.m.f. e = 2000, vary as function of e$ at no load and at a load of i = 100 amp. power current, and what will be the reactive current in the synchronous motor? Let I = ii jiz = current in receiving circuit of voltage e\. Of this current 1,jiz is taken by the synchronous motor of counter e.m.f. 'e, and thus EI = e Zoji 2 = e + X i 2 - jr i 2 ' } or, reduced, e^= (e + xoit) 2 + rjif. In the supply circuit the voltage is Eo = Ei + IZ l = e + xoi* - jr Q i 2 + (ii-jiz) (TI + jxi) = [e + riii + (x Q + xi) i z ] j [(r Q + TI) i 2 ' or, reduced, eo 2 = [e + riti + (X Q + xi) itf + [(r + fi) i 2 Substituting in the equations for e^ and e 2 the above values of r and X Q : at no load, i\ = 0, we have 6l 2 = ( e + 5 z 2 ) 2 + 0.25 i 2 2 and e 2 = (e + at full load, i\ = 100, we have ei 2 = (e + 5 i 2 ) 2 + 0.25 z 2 2 , + = ( e + 50 + 15 - 1000) 2 , 98 ELEMENTS OF ELECTRICAL ENGINEERING and at no load, i\ = 0, substituting e = 2000, we have d 2 = (2000 + 5 i 2 ) 2 + 0.25 z 2 2 , eo 2 = (2000 + 15 * 2 ) 2 + * 2 2 ; at full load, i\ = 100, we have e a 2 = (2000 + 5 * 2 ) 2 + 0.25 * 2 2 , e 2 = (2050 + 15 z 2 ) 2 + fe - 1000) 2 . Substituting herein e = successively 1800, 1900, 2000, 2100, 2200, gives values of i' 2 , which, substituted in the equation for ei 2 , give the corresponding values of ei as recorded in the follow- ing table. As seen, in the local circuit controlled by the synchronous compensator, and separated by reactance from the main circuit of fluctuating voltage, the fluctuations of voltage appear in a greatly reduced magnitude only, and could be entirely eliminated by varying the excitation of the synchronous compensator. e = 2000 No load ii = Full load ' 11 - 100 H ei H e\ 1,800 -13.3 1,937 -39 1,810 1,900 - 6.7 1,965 -30.1 1,850 2,000 2,000 -22 1,885 2,100 + 6.7 2,035 -13.5 1,935 2,200 + 13.3 2,074 - 6.5 1,970 17. IMPEDANCE AND ADMITTANCE 82. In direct-current circuits the most important law is Ohm's law, e -i or e r ir, or r = -.> where e is the e.m.f. impressed upon resistance r to produce current i therein. Since in alternating-current circuits a current i through a resistance r may produce additional e.m.fs. therein, when apply- a ing Ohm's law, i - to alternating-current circuits, e is the IMPEDANCE AND ADMITTANCE ' 99 total e.m.f. resulting from the impressed e.m.f. and all e.m.fs. produced by the current i in the circuit. Such counter e.m.fs. may be due to inductance, as self-induc- tance, or mutual inductance, to capacity, chemical polarization, etc. The counter e.m.f. of self-induction, or e.m.f. generated by the magnetic field produced by the alternating current i, is repre- sented by a quantity of the same dimensions as resistance, and measured in ohms: reactance x. The e.m.f. consumed by reactance x is in quadrature with the current, that consumed by resistance r in phase with the current. Reactance and resistance combined give the impedance, + x 2 ; or, in symbolic or vector representation, Z = r + jx. In general in an alternating-current circuit of current i, the e.m.f. e can be resolved in two components, a power component ei in phase with the current, and a wattless or reactive com- ponent e 2 in quadrature with the current. The quantity e_i _ power e.m.f., or e.m.f. in phase with the current _ i current is called the effective resistance. The quantity 62 _ reactive e.m.f., or e.m.f. in quadrature with the current _ i current is called the effective reactance of the circuit. And the quantity 21 = Vr! 2 + x 2 or, in symbolic representation, Zi = ri + jxi is the impedance of the circuit. If power is consumed in the circuit only by the ohmic resist- ance r, and counter e.m.f. produced only by self-inductance, the effective resistance TI is the true or ohmic resistance r, and the effective reactance Xi is the true or inductive reactance x. 100 ELEMENTS OF ELECTRICAL ENGINEERING By means of the terms effective resistance, effective reactance, and impedance, Ohm's law can be expressed in alternating- current circuits in the form = - e m y / 9 T ~ 9 ; ^ ' Zi vVi 2 + Xi 2 or, e = izi = i V^i 2 + Zi 2 ; (2) or, ! = Vri 8 + a;i a = p ( 3 ) or, in symbolic or vector representation, or, E = IZ l = /(n+jxi); (5) 7^7 or, Zi = ri + jzi = j- (6) In this latter form Ohm's law expresses not only the intensity but also the phase relation of the quantities; thus ei = iri = power component of e.m.f., e z = ix\ = reactive component of e.m.f. p 83. Instead of the term impedance z - with its components, I? the resistance and reactance, its reciprocal can be introduced. e " z ' which is called the admittance. The components of the admittance are called the conduc- tance and the susceptance. Resolving the current i into a power component i\ in phase with the e.m.f. and a wattless component i z in quadrature with the e.m.f., the quantity i\_ _ power current, or current in phase with e.m.f. e e.m.f. . = 9 is called the conductance. The quantity _*2_ _ reactive current, or current in quadrature with e.m.f. e e.m.f. is called the susceptance of the circuit. The conductance represents the current in phase with the IMPEDANCE AND ADMITTANCE 101 e.m.f., or power current, the susceptance the current in quad- rature with the e.m.f., or reactive current. Conductance g and susceptance b combined give the admittance y = Vg 2 + 6 2 ; (7) or, in symbolic or vector representation, Y = g - jb. (8) Thus Ohm's law can also be written in the form i = ey = e Vg 2 + & 2 ; (9) or, i or, y = Vg* + V = 7; (11) or, in symbolic or vector representation, I = EY = E(g-jb); (12) or, E = - or, Y = g - jb = |- (14) and i\ = eg = power component of current, ii = eb = reactive component of current. 84. According to circumstances, sometimes the use of the terms impedance, resistance, reactance, sometimes the use of the terms admittance; conductance, susceptance, is more convenient. Since, in a number of series-connected circuits, the total e.m.f., in symbolic representation, is the sum of the individual e.m.fs., it follows that in a number of series-connected circuits the total impedance, in symbolic expression, is the sum of the impedances of the individual circuits connected in series. Since, in a number of parallel-connected circuits, the total current, in symbolic representation, is the sum of the individual currents, it follows that in a number of parallel-connected cir- cuits the total admittance, in symbolic expression, is the sum of the admittances of the individual circuits connected in parallel. 102 ELEMENTS OF ELECTRICAL ENGINEERING Thus in series connection the use of the term impedance, in parallel connection the use of the term admittance, is generally more convenient. Since in symbolic representation Y = ^ (15) or, ZY = 1; (16) that is, (r+jx)(g - jb) = 1; (17) it follows that (rg + xb) - j (rb - xg) = 1; that is rg -f zb = 1, rb - xg = 0. r = -f- = , (18) 6 = JTqirji = #> (21) or, in absolute values, y = ' (22) (23) (r 2 + x 2 )(^ 2 + 6 2 ) = 1. (24) Thereby the admittance with its components, the conduc- tance and susceptance, can be calculated from the impedance and its components, the resistance and reactance, and inversely. If x = 0, z = r and g = , that is, g is the reciprocal of the resistance in a non-inductive circuit; not so, however, in an inductive circuit. EXAMPLES 85. (1) In a quarter-phase induction motor having an im- pressed e.m.f. e = 110 volts per phase, the current is / = ii jiz = 100 100 j at standstill, the torque = D . The two phases are connected in series in a single-phase cir- cuit of e.m.f. e = 220, and one phase shunted by a condenser of 1 ohm capacity reactance. What is the starting torque D of the motor under these con- ditions, compared with Z> , the torque on a quarter-phase cir- IMPEDANCE AND ADMITTANCE 103 cuit, and what the relative torque per volt-ampere input, if the torque is proportional to the product of the e.m.fs. impressed upon the two circuits and the sine of the angle of phase dis- placement between them? In the quarter-phase motor the torque is D = ae 2 = 12,100 a, where a is a constant. The volt-ampere input is Qo = 2 e Vii 2 + i 2 2 = 31,200; hence, the "apparent torque efficiency," or torque per volt- ampere input, rj Q = D* = 0.388 a. The admittance per motor circuit is the impedance is Y = = 0.91 - 0.91 j, e _ 110 (100 + 100 j) _055+055/ I ~ 100- 100 j ~ (100.-100j)(100+100j)~ the admittance of the condenser is Yo = j; thus, the joint admittance of the circuit shunted by the con- denser is Yi= Y + 7o = 0.91 - 0.91 j + j = 0.91 +0.09 j; its impedance is 7 J_ _ L_ 0.91- 0.09 j Zl ~ F, ~ 0.91 + 0.09 j ~ 0.9P + 0.09 2 = X 3 > and the total impedance of the two circuits in series is Z 2 = Z + Z l = 0.55 + 0.55 j + 1.09 - 0.11 j = 1. 64 + 0.44 j. Hence, the current, at impressed e.m.f. e = 220. r . .. e 220 220 (1.64- 0.44 j) !ti ^ 2 ~ Z 2 ~ 1.64 + 0.44 j~ 1.64 2 + = 125 - 33.5 j; 104 ELEMENTS OF ELECTRICAL ENGINEERING or, reduced, / = V125 2 + 33.5 2 = 129.4 amp. Thus, the volt-ampere input, Q = el = 220 X 129.4 = 28,470. The e.m.fs. acting upon the two motor circuits respectively are Ei = /Zi = (125 - 33.5 j) (1.09 - 0.11 j) = 132.8 - 50.4 j and E' = IZ = (125 - 33.5 j) (0.55 + 0.55 j) = 87.2 + 50.4 j. Thus, the tangents of their phase angles are 50 4 tan 0i = + - = + 0.30; hence, B l = + 21; 50 4 tan tf = - = - 0.579; hence, 6' = - 30; and the phase difference, = 0i The absolute values of these e.m.fs. are ' = 51. ei = x/132.8 + 50.4 2 = 141.5 and e' = V87.2 2 - 50.4 2 = 100.7; thus, the torque is D = ae\e' sin 6 = 11, 100 a; and the apparent torque efficiency is _D 11,100 a " ~ Q WTO" Hence, comparing this with the quarter-phase motor, the relative torque is D_ = 11,100 a Do 12,100 a and the relative torque per volt-ampere, or relative apparent torque efficiency, is it 0.39 a 0.388 a = 1.005. IMPEDANCE AND ADMITTANCE 105 86. (2) At constant field excitation, corresponding to a nominal generated e.m.f. JS Q 12,000, a generator of synchro- nous impedance Z Q = r + J^o = 0.6 + 60 j feeds over a trans- mission line of impedance Z\ = ri + jx\ = 12 '+ 18 j, and of capacity susceptance 0.003, a non-inductive receiving circuit. How will the voltage at the receiving end, e, and the voltage at the generator terminals, e\, vary with the load if the line capacity is represented by a condenser shunted across the middle of the line? Let I = i = current in receiving circuit, in phase with the e.m.f., E = e. The voltage in the middle of the line is = e + 6 i + 9 ij. The capacity susceptance of the line is, in symbolic expression, Y = 0.003 j; thus the charging current is 7 2 = E 2 Y = 0.003 j (e + 6 i + 9 ij) = 0.027 i + j (0.003 e + 0.018 i), and the total current is /! = I + 7 2 = 0.973 i + j (0.003 e + 0.018 i). Thus, the voltage at the generator end of the line is = e + 6 i + 9 ij + (6 + 9 j)[0.973 i + j (0.003 e + 0.018 i)] = (0.973 * + 11.68 i) + j (17.87 i + 0.018 e), and the nominal generated e.m.f. of the generator is E = E! + Zo|i = (0.973 e + 11.68 i) + j (17.87 i + 0.018 e) + (0.6 + 60 j) [0.973 t + j (0.003 e + 0.018 t)] = (0.793 e + 11.18 i) + j (76.26 i + 0.02 e); or, reduced, and e = 12,000 substituted, e 2 = 144 x 10 6 = (0.793 e + 11.18 i) 2 + (76.26 i + 0.02 e) 2 ; thus, e 2 + 33 ei + 9450 i 2 = 229 X 10 6 , e = - 16.5 i + V229 X 10 6 - 9178 1*, 106 ELEMENTS OF ELECTRICAL ENGINEERING and ei = V(0.973e + 11.68 *) 2 -f (17.87 i + O.OlSe) 2 ; at i = 0, e = 15,133, a = 14,700; at e = 0, i = 155.6, d = 3327. P )WER CURRENT REC'D AMP V *OLT8 5000 3000 5000 3000 2000 10 20 JO 40 50 DO 70 80 90 100 110 120 130 140 150 FIG. 40. Reactive load characteristics of a transmission line fed by synchronous generator with constant field excitation. Substituting different values for i gives i ' ei i e ei 15,133 14,700 100 10,050 11,100 25 14,488 14,400 125 7,188 8,800 50 13,525 13,800 150 2,325 4,840 75 12,063 12,730 155.6 3,327 which values are plotted in Fig. 40. 18. EQUIVALENT SINE WAVES 87. In the preceding chapters, alternating waves have been assumed and considered as sine waves. EQUIVALENT SINE WAVES 107 The general alternating wave is, however, never completely, frequently not even approximately, a sine wave. A sine wave having the same effective value, that is, the same square root of mean squares of instantaneous values, as a general alternating wave, is called its corresponding "equivalent sine wave." It represents the same effect as the general wave. With two alternating waves of different shapes, the phase relation or angle of lag is indefinite. Their equivalent sine waves, however, have a definite phase relation, that which gives the same effect as the general wave, that is, the same mean (ei). Hence if e = e.m.f. and i = current of a general alternating wave, their equivalent sine waves are defined by e = -\Anean (e 2 ), io = A/mean (i 2 ); and the power is Po = e Q iQ cos eoiQ = mean (ei)', thus, mean (ei) COS QIQ = / - Vmean (e 2 ) v mean (i 2 ) Since by definition the equivalent sine waves of the general alternating waves have the same effective value or intensity and the same power or effect, it follows that in regard to inten- sity and effect the general alternating waves can be represented by their equivalent sine waves. Considering in the preceding the alternating currents as equiva- lent sine waves representing general alternating waves, the investigation becomes applicable to any alternating circuit irrespective of the wave shape. The use of the terms reactance, impedance, etc., implies that a wave is a sine wave or represented by an equivalent sine wave. Practically all measuring instruments of alternating waves (with exception of instantaneous methods) as ammsters, volt- meters, wattmeters, etc., give not general alternating waves but their corresponding equivalent sine waves. EXAMPLES 88. In a 25-cycle alternating-current transformer, at 1000 volts primary impressed e.m.f., of a wave shape as shown in 108 ELEMENTS OF ELECTRICAL ENGINEERING e M OCOOI>.C^O5(NCOOOOi' l i 1 CO CO CO H i 1 OQ '^ CO CO C^J ^H >O CO iQ CO C^ O O5 CO CO iQ CO O TfitOOOiOiOO ^ rH 00 00 T-( c co" co"co"co' N co > i-r ^^^ooooooo 1 O .. 00 00^2 ^ ^ ^ se ^ oo oo . < r-l CO ?CI> i-H QJ CO EQUIVALENT SINE WAVES 109 Fig. 41 and Table I, the number of primary turns is 500, the length of the magnetic circuit 50 cm., and its section shall be chosen so as to give a maximum density B = 15,000. At this density the hysteretic cycle is as shown in Fig. 42 and Table II. FIG. 41. Wave-shape of e.m.f. in example 88. What is the shape of current wave, and what the equivalent sine waves of e.m.f., magnetism, and current? The calculation is carried out in attached table. TABLE II / B 8 ,000 2 + 10,400 - 2,500 4 + 11,700 + 5,800 6 + 12,400 + 9,300 8 + 13,000 + 11,200 10 + 13,500 + 12,400 12 + 13,900 + 13,200 14 + 14,200 + 13,800 16 + 14,500 + 14,300 18 + 14,800 + 14,700 20 + 15,000 In column (1) are given the degrees, in column (2) the relative values of instantaneous e.m.fs., e corresponding thereto, as taken from Fig. 41. Column (3) gives the squares of e. Their sum is 24,939; 24 939 thus the mean square, . ' * - = 1385.5, and the effective value, lo 110 ELEMENTS OF ELECTRICAL ENGINEERING Since the effective value of impressed e.m.f. is = 1000, the 1 000 instantaneous values are e Q = e^-^ as given in column (4). Since the e.m.f. e is proportional to the rate of change of magnetic flux, that is, to the differential coefficient of B } B is proportional to the integral of the e.m.f., that is, to Se plus an integration constant. 2e is given in column (5), and the integration constant follows from the condition that B at 180 FIG. 42. Hysteretic cycle in example 88. must be equal, but opposite in sign, to B at 0. The integration constant is, therefore, 1 SO -it n i ^ and by subtracting 7324 from the values in column (5) the values of B' of column (6) are found as the relative instantaneous values of magnetic flux density. Since the maximum magnetic flux density is 15,000 the in- 15 000 stantaneous values are B = B' ' . , plotted in column (7). From the hysteresis cycle in Fig. 42 are taken the values of magnetizing force /, corresponding to magnetic flux density B. They are recorded in column (8), and in column (9) the instan- taneous values of m.m.f. F = If, where I = 50 = length of magnetic circuit. EQUIVALENT SINE WAVES 111 i = , where n = 500 = number of turns of the electric circuit, gives thus the exciting current in column (10) . Column (11) gives the squares of the exciting current, i 2 . 25 85 Their sum is 25.85; thus the mean square, ' = 1.436, and lo the effective value of exciting current, i' = Vl.436 = 1.198 amp. Column (12) gives the instantaneous values of power, p = ieo. Their sum is 4766; thus the mean power, p' = 4766 18 = 264.8. FIG. 43. Waves of exciting current. Power and flux density corresponding to e.m.f . in Fig. 41 and hysteretic cycle in Fig. 42. FIG. 44. Corresponding sine waves for e.m.f. and exciting current in Fig. 43. Since p' = i'e' Q cos 0, where e' and i' are the equivalent sine waves of e.m.f. and of current respectively, and their phase displacement, substitut- ing these numerical values of p', e r , and i', we have 264.8 = 1000 X 1.198 cos 6. hence, cos = 0.221, 6 = 77.2, 112 ELEMENTS OF ELECTRICAL ENGINEERING and the angle of hysteretic advance of phase, a = 90 - = 12.8. The hysteresis current is then i' cos e = 0.265, and the magnetizing current, i' sin = 1.165. Adding the instantaneous values of e.m.f. e Q in column (4) 14 648 gives 14,648; thus the mean value, f-r = 813.8. Since the J-O effective value is 1000, the mean value of a sine wave would be 2 -v/2 1000- - = 904; hence the form factor is 7T 904 7 = Adding the instantaneous values of current i in column (10), irrespective of their sign, gives 17.17; thus the mean value, 17.17 ' = 0.954. Since the effective value = 1.198, the form lo factor is 1.198 2 V2 7 = 0954 The instantaneous values of e.m.f. e , current i, flux density B and power p are plotted in Fig. 43, their corresponding sine waves in Fig. 44. 19. FIELDS OF FORCE 89. When an electric current flows through a conductor, power is consumed and heat produced inside of the conductor. In the space outside and surrounding the conductor, a change has taken place also, and this space is not neutral and inert any more, but if we try to move a solid mass of metal rapidly through it, the motion is resisted, and heat produced in the metal by induced currents. Materials of high permeability, as iron filings, brought into this space arrange themselves in chains; a magnetic needle is moved and places itself in a definite direction. Due to the passage of the current in the conductor, there are therefore in the spaces outside of the con- ductor where the current does not flow forces exerted, and FIELDS OF FORCE 113 this space then is not neutral space, but has become a field of force, and the cause of the field, in this case the electric current in the conductor, is its "motive force." As in this case the actions exerted in the field of force are magnetic, the space surrounding a conductor traversed by a current is a field of magnetic force, and the current in the conductor is the magneto- motive force. In the space surrounding a ponderable mass, as our earth, forces are exerted on other masses which cause the stone to fall toward the earth, and water to run down hill and this space thus is a field of gravitational force, the earth the gram- motive force. In the space surrounding conductors having a high potential difference, we observe a field of dielectric force, that is, electro- static or dielectric forces are exerted, and the potential difference between the conductors is the electromotive force of the dielectric field. The force exerted by the earth as gravimotive force, on any mass in the gravitational field of the earth, causes the mass to move with increasing rapidity. The direction of motion then shows the direction in which the force acts, that is, the direc- tion of the gravitational field. The force g, which the field exerts on unit mass, that is, the acceleration of the mass, measures the intensity of the field: in the gravitational field of the earth 981 cm g sec. The force acting upon a mass m, then, is:F = gm, and is called the weight of the mass. In the same manner, in the magnetic field of a current as magnetomotive force, the intensity H of the magnetic field is measured by the force F which the field exerts on a magnetic mass or pole strength m: F = Hm; the intensity K of the di- electric field of a potential difference as electromotive force is measured by the force F exerted upon an electric pole strength e: F = Ke', the direction of the force represents the direction of the field of force. 90. This conception of the field of force is one of the most important and fundamental ones of all sciences and applied sciences: a condition of space, brought about by some exciting cause or motive force, whereby the space is not neutral any more, but capable of exerting forces on anything susceptible to these forces: mechanical forces on masses in a gravitational field, magnetic forces on magnetic materials in a magnetic field, 114 ELEMENTS OF ELECTRICAL ENGINEERING A. A photograph of a mica-filing map of the dielectric lines of force- between two cylinders. B. A photograph of an iron-filing map of the magnetic lines of force about. two cylinders. C. A photographic superposition of A and B representing the magnetic- and dielectric fields of the space surrounding two conductors which are; carrying energy. FIG. 45. FIELDS OF FORCE 115 dielectric forces on dielectrics in a dielectric field, etc. The field of force then is characterized by having, at any point, a definite direction the direction in which the force acts and a definite intensity, to which the forces are proportional. Such fields of force can be graphically represented by lines showing the direction in which the force acts: the lines of force and, at right angles thereto, the equipotential lines or surfaces, as the direction in which no force acts. Thus the lines of gravita- tional force of the earth are the verticals, the equipotential sur- faces, or level surfaces, are the horizontals. Such pictures of a field of force also illustrate the intensity: where the lines of force and therefore the equipotential lines come closer together, the field is more intense, that is, the forces greater. FIG. 46. A mathematical plot of fields shown in C. Magnetic fields may be demonstrated by iron filings brought into the field; dielectric fields by particles of a material of high specific capacity, such as mica. Fig. 45 shows the dielectric field of a pair of parallel conductors, the magnetic field between these conductors, and their combination. Fig. 46 shows the same as calculated. As further illustration, Fig. 47 shows, from observation, half of the dielectric field between a rod with circular disc, as one terminal, passing symmetrically through the center of a cylinder placed in a circular hole in a plate as other terminal: the lines of force pass from terminal to terminal; the equipotential surfaces intersect at right angles (A 10,292). 91. In electrical engineering we have to deal with the electrical quantities: voltage, current, resistance, etc.; the magnetic quan- 116 ELEMENTS OF ELECTRICAL ENGINEERING titles: magnetic flux, field intensity, permeability, etc.; and the di- electric quantities: dielectric flux, field intensity, permittivity, etc. The electric current is the magnetomotive force F which produces the magnetic field, acting upon space. It is expressed in amperes, or rather in ampere-turns, and thus is an electrical quantity, its Rod V-0 Plane Hole FIG. 47. Observed dielectric field. unit being determined by the unit of current, as the ampere-turn equal to 10" 1 absolute units. The magnetomotive force per unit length of the magnetic circuit then is the magnetizing force or magnetic gradient f, in ampere-turns per centimeter, hence still an electrical quantity. Proportional thereto, and of the same dimension, is the FIELDS OF FORCE 117 magnetic field intensity H. It differs from the magnetic gradient merely by a numerical factor 4 TT; H = 4?r/ 10" 1 . Magnetic field intensity is a magnetic quantity, and its unit defined by the magnetic forces exerted in the field, thus different from the unit of magnetic gradient, which is determined by the unit of electric current; hence the factor 4 IT. The factor 10" 1 merely reduces from amperes to absolute unit. If then v is the magnetic conductivity of the material in the magnetic field, called its permeability, B = pH is the magnetic flux density, and the total magnetic flux <1> is given by the density B times the area or section of the flux. Or, passing directly from the magnetomotive force F to the F magnetic flux, by the conception of the magnetic circuit: 3> = > where R is the magnetic resistance, or reluctance of the magnetic circuit. R is an electric quantity, and does not contain the 4 TT. In the dielectric field, the potential difference e is the electro- motive force expressed in volts. The electromotive force per unit length of the dielectric circuit is the electrifying force or voltage gradient or dielectric gradient g, expressed in volts per centimeter. This is still an electric quantity. Proportional thereto by a numerical factor is the dielectric quantity: dielectric field intensity K = . 2 , and if k is the dielectric conductivity of the medium in the dielectric field, called specific capacity or permittivity, the dielectric flux density is D = kK, and the total dielectric flux ^ is flux density times area. Here again, at the transition from the electric quantity "gradient" to the dielectric quantity " field intensity," a numer- ical factor 4 irv 2 enters, the one quantity being based on the volt as unit, the other on unit force action, v is the velocity of light, 3 X 10 10 , and the factor v 2 the result of the convention of assum- ing the permittivity of empty space as unity. It is now easy to remember, where in the electromagnetic system of units the factor 4-Tr enters: it is at the transition from the electrical quantities to the magnetic or dielectric quantities, from gradient to field intensity. 92. The dielectric field and the magnetic field are analogous, and to magnetic flux, magnetic field intensity, permeability, as used in dealing with magnetic circuits, correspond the terms 118 ELEMENTS OF ELECTRICAL ENGINEERING dielectric flux, dielectric field intensity, permittivity, as used in dealing with the electrostatic fields of high potential apparatus, as transmission insulators, transformer bushings, etc. The fore- most difference is that in the magnetic field, a line of force must always return into itself in a closed circuit, while in the electro- static or dielectric field, a line of force may terminate in a con- ductor. The terminals of the lines of electrostatic flux, ^ at the conductor, then are represented by the conception of a quantity of electricity or electric charge, Q, being located on the con- ductor. Thus, at the terminal of the line of unit dielectric flux, unit electric quantity is located on the conductor. Dielectric flux ^ and electric quantity or charge Q thus are identical, and merely different conceptions of the dielectric circuit : Q = *. In using the conception of electric quantity Q, we consider only the terminals of the lines of dielectric flux, that is, deal merely with the effect of the dielectric flux on the electric circuit which produced it. This conception is in many cases more convenient, but it necessarily fails, when the distribution of the dielectric flux in the dielectric field is of importance, such as is the case when dealing with high dielectric field intensities, approach- ing the possibility of disruptive effects in the field of force, or when dealing with the effect produced by the introduction of ma- terials of different permittivity into the dielectric field. There- fore, with the increasing importance of the dielectric field in engineering, the conception of electric quantity, or charge, is gradually being replaced by the conception of the dielectric flux and the dielectric field, analogous to the magnetic field, which has replaced the previous conception of " magnetic poles." 20. NOMENCLATURE 93. The following nomenclature and symbols of the quantities most frequently used in electrical engineering appears most satisfactory, and is therefore recommended. It is in agreement with the Standardization Rules of the A. I. E. E., but as far as possible standard letters have been used, and script letters avoided as impracticable or at least inconvenient in writing and still more in typewriting. Therefore F has been chosen for m.m.f., and dielectric field intensity changed to K. Also, a few symbols not contained in the Standardization Rules had to be added. NOMENCLATURE TABLE OP SYMBOLS 119 Symbol Name Unit Character E, e. Voltage Volt Electrical I, i. . Potential difference Electromotive force Current Ampere Electrical R,r Resistance Ohm Electrical x Reactance Ohm Electrical Z,z... Impedance Ohm Electrical a Conductance Mho Electrical b Susceptance Mho Electrical Y,y p Admittance Resistivity Mho Ohm-centimeter Electrical Electrical 7 $ Conductivity Magnetic flux Mho-centimeter Line; kiloline; megaline Electrical Magnetic ....... H Magnetic density Magnetic field inten- Lines per cm. 2 ; kilo- lines per cm. 2 Lines per cm 2 Magnetic Magnetic /* sity Permeability (magnetic Magnetic / conductivity) Magnetic gradient Ampere-turns per centi- Electrical F Magnetizing force Magnetomotive force meter. Ampere-turns Electrical R Reluctance (magnetic Electrical L M S .. . resistance) Inductance Mutual inductance Self-inductance Henry; milhenry Henry; milhenry Henry; milhenry Magnetic Magnetic Magnetic *,Q D K Leakage inductance Dielectric flux Electric quantity or charge Dielectric density Dielectric field inten- Lines of dielectric force Coulombs Dielectric lines per cm. 2 Coulombs per cm. 2 Dielectric Dielectric Dielectric k sity Permittivity Dielectric Specific capacity 120 ELEMENTS OF ELECTRICAL ENGINEERING TABLE OF SYMBOLS. Continued Symbol Name Unit Character 9 Dielectric gradient Volts per centimeter Electrical Voltage gradient Electrifying force C Capacity Farad; microfarad Dielectric P,P Power, effect Watt; kilowatt General W,w.... Energy, work Joule; kilo joule General T,# Temperature Degrees Centigrade General t Time Seconds General $,$,0... Time angle Degrees or radians General <*,T Space angle Degrees or radians General / Frequency Cycles per second General PART II SPECIAL APPARATUS INTRODUCTION 1. By the direction of the energy transmitted, electric machines have been divided into generators and motors. By the character of the electric power they have been distinguished as direct- current and as alternating-current apparatus. With the advance of electrical engineering, however, these subdivisions have become unsatisfactory and insufficient. The division into generators and motors is not based on any characteristic feature of the apparatus, and is thus not rational. Practically any electric generator can be used as motor, and conversely, and frequently one and the same machine is used for either purpose. Where a difference is made in the construction, it is either only quantitative, as, for instance, in synchronous motors a higher armature reaction is often used than in synchro- nous generators, or it is in minor features, as direct-current motors usually have only one field winding, either shunt or series, while in generators frequently a compound field is employed. Further- more, apparatus have been introduced which are neither motors nor generators, as the synchronous machine producing wattless lag- ging or leading current, etc., and the different types of converters. The subdivision into direct-current and alternating-current apparatus is unsatisfactory, since it includes in the same class apparatus of entirely different character, as the induction motor and the alternating-current generator, or the constant-potential commutating machine and the rectifying arc light machine. Thus the following classification, based on the characteristic features of the apparatus, as adopted by the A. I. E. E. Standard- izing Committee, is used in the following discussion. It refers only to the apparatus transforming between electric and electric and between electric and mechanical power. 1st. Commutating machines, consisting of a magnetic field and a closed-coil armature, connected with a multi-segmental commutator. 121 122 ELEMENTS OF ELECTRICAL ENGINEERING 2d. Synchronous machines, consisting of a undirectional mag- netic field and an armature revolving relatively to the mag- netic field at a velocity synchronous with the frequency of the alternating-current circuit connected thereto. . 3d. Rectifying apparatus, that is, apparatus reversing the direc- tion of an alternating current synchronously with the frequency. 4th. Induction machines, consisting of an alternating mag- netic circuit or circuits interlinked with two electric circuits or sets of circuits moving with regard to each other. 5th. Stationary induction apparatus, consisting of a magnetic circuit interlinked with one or more electric circuits. 6th. Electrostatic and electrolytic apparatus as condensers and polarization cells. Apparatus changing from one to a different form of electric energy have been defined as: A. Transformers, when using magnetism, and as B. Converters, when using mechanical momentum as inter- mediary form of energy. The transformers as a rule are stationary, the converters rotary apparatus. Motor-generators transforming from elec- trical over mechanical to electric power by two separate machines, and dynamotors, in which these two machines are combined in the same structure, are not included under converters. 2. (1) Direct-current commutating machines as generators are Usually built to produce constant potential for railway, incan- descent lighting, and general distribution. As motors commutat- ing machines give approximately constant speed shunt motors or large starting torque series motors. When inserted in series in a circuit, and controlled so as to give an e.m.f. varying with the conditions of load on the system, these machines are "boosters," and are generators when raising the voltage, and motors when lowering it. Commutating machines may be used as direct-current con- verters by transforming power from one side to the other side of a three- wire system. Alternating-current commutating machines are used as motors of series characteristic for railway and other varying speed service, or with shunt characteristic for constant speed and adjustable speed work, especially where high starting torque efficiency is required. They usually are of single-phase type. (2) While in commutating machines the magnetic field is, INTRODUCTION 123 almost always stationary and the armature rotating, synchronous machines were built with stationary field and revolving armature, or with stationary armature and revolving field, or as inductor machines with stationary armature and stationary field winding but revolving magnetic circuit. Generally now the revolving field type is used. By the number and character of the alternating circuits con- nected to them they are single-phase or polyphase machines. As generators they comprise practically all single-phase and poly- phase alternating-current generators; as motors a very important class of apparatus, the synchronous motors, which are usually preferred for large powers, especially where frequent starting and considerable starting torque are not needed. Synchronous machines may be used as compensators or synchronous condensers, to produce wattless current, leading by over-excitation, lagging by under-excitation, or may be used as phase converters by operat- ing a polyphase synchronous motor by one pair of terminals from a single-phase circuit. The most important class of converters, however, are the synchronous commutating machines, to which, therefore, a special chapter will be devoted in the following. Inserted in series to another synchronous machine or synchro- nous converter, and rigidly connected thereto, synchronous ma- chines are also occasionally used as boosters. Synchronous commutating machines contain a unidirectional magnetic field and a closed circuit armature connected simul- taneously to a segmental direct-current commutator and by collector rings to an alternating circuit, generally a polyphase system. Thus these machines can either receive alternating and yield direct-current power as synchronous converters or simply " converters," or receive direct and yield alternating-current power as inverted converters, or driven by mechanical power yield alternating and direct current as double-current generators. Or they can combine motor and generator action with their converter action. Thus a combination is a synchronous con- verter supplying a certain amount of mechanical power as a synchronous motor. Usually, they convert from three-phase or single-phase alternating to direct-current power. (3) Rectifying machines are apparatus which by a synchro- nously revolving rectifying commutator send the successive half waves of an alternating single-phase or polyphase circuit in the same direction into the receiving circuit. The most impor- 124 ELEMENTS OF ELECTRICAL ENGINEERING tant class of such apparatus were the open-coil arc light ma- chines. They have been practically superseded by the mercury arc rectifier. (4) Induction machines are generally used as motors, poly- phase or single-phase. In this case they run at practically constant speed, slowing down slightly with increasing load. As generators the frequency of the e.m.f. supplied by them differs from and is lower than the frequency of rotation, but their opera- tion depends upon the phase relation of the external circuit. As phase converters, induction machines can be used in the same manner as synchronous machines. Another occasional use be- sides as motors is, however, as frequency converters, by changing from an impressed primary polyphase system to a secondary polyphase system of different frequency. In this case, when low- ering the frequency, mechanical energy is also produced; when raising the frequency, mechanical energy is consumed. (5) The most important stationary induction apparatus is the transformer, consisting of two electric circuits interlinked with the same magnetic circuit. When using the same or part of the same electric circuit for primary and secondary, the transformer is called an auto-transformer or compensator. When inserted in series into a circuit, and arranged to vary the e.m.f., the trans- former is called potential regulator or booster. The variation of secondary e.m.f. may be secured by varying the relative number of primary and secondary turns, or by varying the mutual in- ductance between primary and secondary circuit, either elec- trically or magnetically. The stationary induction apparatus with one electric circuit are used for producing wattless lagging currents, as reactors, reactive or choke coils. (6) Condensers and polarization cells produce wattless leading currents, the latter, however, usually at a low efficiency, while the efficiency of the condenser is extremely high, frequently above 99 per cent. ; that is, the loss of power is less than 1 per cent, of the apparent volt-ampere input. Unipolar, or, more correctly, non-polar or acyclic machines are apparatus in which a conductor cuts a continuous magnetic field at a uniform rate. They have not become of industrial importance. Regarding apparatus transforming between electric energy and forms of energy differing from electric or mechanical energy: The transformation between electrical and chemical energy is INTRODUCTION 125 represented by the primary and secondary battery and the elec- trolytic cell; the transformation between electrical and heat energy by the thermopile and the electric heater or electric fur- nace; the transformation between electrical and light energy by the incandescent and arc lamps. In the following will be given a general discussion of the charac- teristics of the most frequently used and therefore most impor- tant classes of apparatus. A further discussion and calculation of these apparatus is given in "Theory and Calculation of Alternating Current Phenomena," while a discussion of those characteristics and modifications of these apparatus, which, though important, are less frequently met, and a discussion of the numerous less common types of apparatus, which could not be included in the following, is given in "Theory and Calculation of Electrical Apparatus." Some important features, as the nature of the reactance of apparatus, mechanical magnetic forces, wave shape distortions caused by some features of design, in apparatus, etc., are dis- cussed in "Theory and Calculation of Electric Circuits." A. SYNCHRONOUS MACHINES I. General 3. The most important class of alternating-current apparatus consists of the synchronous machines. They comprise the alternating-current generators, single-phase and polyphase, the synchronous motors, the phase compensators, the phase con- verters, the phase balancers, the synchronous boosters and the exciters of induction generators, that is, synchronous machines producing wattless lagging or leading currents, and the con- verters. Since the latter combine features of the commutating machines with those of the synchronous machines they will be considered separately. In the synchronous machines the terminal voltage and the generated e.m.f. are in synchronism with, that is, of the same frequency as, the speed of rotation. These machines consist of an armature, in which e.m.f. is generated by the rotation relatively to a magnetic field, and a continuous magnetic field, excited either by direct current, or by the reaction of displaced phase armature currents, or by per- manent magnetism. The formula for the e.m.f. generated in synchronous machines, commonly called alternators, is E = S2irn3> = 4. where n is the number of armature turns in series interlinked with the magnetic flux <1> (in megalines per pole), / the frequency of rotation (in hundreds of cycles per second), E the e.m.f. gen- erated in the armature turns. This formula assumes a sine wave of e.m.f. If the e.m.f. wave differs from sine shape, the e.m.f. is E = 4.447/n, 2 -\/2 where y = form factor of the wave, or - times ratio of effect- 7T ive to mean value of wave, that is, the ratio ,of the effective value of the generated e.m.f. to that of a sine wave generated by the same magnetic flux at the same frequency. 126 SYNCHRONOUS MACHINES 127 The form factor 7 depends upon the wave shape of the gener- ated e.m.f. The wave shape of e.m.f. generated in a single con- ductor on the armature surface is identical with that of the dis- tribution of magnetic flux at the armature surface and will be discussed more fully in the chapter on commutating machines. The wave of total e.m.f. is the sum of the waves of e.m.f. in the individual conductors, added in their proper phase relation, as corresponding to their relative positions on the armature surface. 4. In a Y or star-connected three-phase machine, if E Q = e.m.f. per circuit, or Y or star e.m.f., E = E \/3 is the e.m.f. between terminals or A (delta) or ring e.m.f., since two e.m.fs. displaced by 60 degrees are connected in series between terminals (V3 = 2 cos 30). In a A-connected three-phase machine, the e.m.f. per circuit is the e.m.f. between the terminals, or A e.m.f. In a F-connected three-phase machine, the current per circuit is the current issuing from each terminal, or the line current, or Y current. In a A-connected three-pHase machine, if J = current per circuit, or A current, the current issuing from each terminal, or the line or F current, is / = /o V3. Thus in a three-phase system, A current and e.m.f., and F current and e.m.f. (or ring and start current and e.m.f. respect- ively), are to be distinguished. They stand in the proportion 1 - V3. As a rule, when speaking of current and of e.m.f. in a three- phase system, under current the F current or current per line, and under e.m.f. the A e.m.f. or e.m.f. between lines is understood. 5. While the voltage wave of a single conductor has the same shape as the distribution of the magnetic flux at the armature circumference and so may differ considerably from a sine, that is, contain pronounced higher harmonics, the terminal voltage is the resultant of the waves of many conductors, and, especially with a distributed armature winding, shows the higher harmonics in a much reduced degree; that is, the resultant is nearer sine shape, and some harmonics may be entirely eliminated in the terminal voltage wave, though they may appear in the voltage wave of a single conductor. Thus, for instance, in a three-phase F-connected machine, the voltage per circuit, or F voltage, may contain a third harmonic and multiples thereof, while in the 128 ELEMENTS OF ELECTRICAL ENGINEERING voltage between the terminals this third harmonic is eliminated. The voltage between the terminals is the resultant of two Y voltages, displaced from each other by 60 degrees. Sixty de- grees for the fundamental, however, is 3 X 60 = 180, or oppo- sition for the third harmonic; that is, the third harmonics in those two Y voltages, which combine to the delta or terminal voltage, are opposite, and so neutralize each other. Even in a single turn, harmonics existing in the magnetic field and thus in the single conductor can be eliminated by fractional pitch. Thus, if the pitch of the armature turn is not 180 de- grees, but less by -> the e.m.fs. generated in the two conductors n of a single turn are not exactly in phase, but differ by - of a half fl> wave for the fundamental, and thus a whole half wave for the nth harmonic, so that their nth harmonics are in opposition and thus cancel. Fractional pitch winding of a "pitch deficiency" of - thus eliminates the nth harmonic; for instance, with 80 per ^ . cent, pitch, the fifth harmonic cannot exist. In this manner higher harmonics of the e.m.f. wave can be reduced or entirely eliminated, though in general, with a dis- tributed winding, the wave shape is sufficiently close to sine shape without special precaution being taken in the design. II. Electromotive Forces 6. In a synchronous machine we have to distinguish between terminal voltage E, real generated e.m.f. #1, virtual generated e.m.f. EZ, and nominal generated e.m.f. E Q . The real generated e.m.f. EI is the e.m.f. generated in the alter- nator armature turns by the resultant magnetic flux, or mag- netic flux interlinked with them, that is, by the magnetic flux passing through the armature core. It is equal to the terminal voltage plus the e.m.f. consumed by the resistance of the arma- ture, these two e.m.fs. being taken in their proper phase relation; thus Ei = E + Ir, where / = current in armature, r = effective resistance. The virtual generated e.m.f. E 2 is the e.m.f. which would be generated by the flux produced by the field poles, or flux corre- sponding to the resultant m.m.f., that is, the resultant of the SYNCHRONOUS MACHINES 129 m.m.fs. of field excitation and of armature reaction. Since the magnetic flux produced by the armature, or flux of armature self-inductance, combines with the field flux to the resultant flux, the flux produced by the field poles does not pass through the armature completely, and the virtual e.m.f. and the real gener- ated e.m.f. differ from each other by the e.m.f. of armature self- inductance; but the virtual generated e.m.f., as well as the e.m.f. generated in the armature by self-inductance, have no real and independent existence, but are merely fictitious components of the real or resultant generated e.m.f. EI. The virtual generated e.m.f. is Ei = Et + jlx, where x is the self -inductive armature reactance, and the e.m.f consumed by self-inductance Ix is to be combined with the real generated e.m.f. EI in the proper phase relation. 7. The nominal generated e.m.f. E Q is the e.m.f. which would be generated by the field excitation if there were neither self- inductance nor armature reaction, and the saturation were the same as corresponds to the real generated e.m.f. It thus does not correspond to any magnetic flux, and has no existence at all, but is merely a fictitious quantity, which, however, is very useful for the investigation of alternators by allowing the combination of armature reaction and self-inductance into a single effect by a (fictitious) self-inductance or synchronous reactance XQ. The nominal generated e.m.f. would be the terminal voltage with open circuit and load excitation if the saturation curve were a straight line. The synchronous reactance XQ is thus a quantity combining armature reaction and self-inductance of the alternator. It is the only quantity which can easily be determined by experiment by running the alternator on short circuit with excited field. If in this case IQ = current, PQ = loss of power in the armature coils, EQ = e.m.f. corresponding to the field excitation at open w p circuit, 7 = Z Q is the synchronous impedance, y^ = r is the -to J-o effective resistance (ohmic resistance plus load losses), and XQ = A/2 2 r o 2 the synchronous reactance. In this feature lies the importance of the term " nominal generated e.m.f." E Q , E = Ei + J!XQ, = E + (r + jx) I 130 ELEMENTS OF ELECTRICAL ENGINEERING the terms being combined in their proper phase relation. In a polyphase machine, these considerations apply to each of the machine circuits individually. III. Armature Reaction 8. The magnetic flux in the field of an alternator under load is produced by the resultant m.m.f. of the field exciting current and of the armature current. It depends upon the phase rela- tion of the armature current. The e.m.f. generated by the field exciting current or the nominal generated e.m.f. reaches a maxi- mum when the armature coil faces the position midway between FIG. 48. Model for study of armature reaction. Armature coils in position of maximum current. the field poles, as shown in Fig. 48, A and A'. Thus, if the armature current is in phase with the nominal generated e.m.f., it reaches its maximum in the same position A, A' of armature coil as the nominal generated e.m.f., and thus magnetizes the preceding, demagnetizes the following magnet pole (in the di- rection of rotation) in an. alternating-current generator A] magnetizes the following and demagnetizes the preceding mag- net pole in a synchronous motor A' (since in a generator the rotation is against, in a synchronous motor with the magnetic attractions and repulsions between field and armature). In this case the armature current neither magnetizes nor demag- netizes the field as a whole, but magnetizes the one side, demag- SYNCHRONOUS MACHINES 131 netizes the other side of each field pole, and thus merely distorts the magnetic field. 9. If the armature current lags behind the nominal generated e.m.f., it reaches its maximum in a position where the armature coil already faces the next magnetic pole, as shown in Fig. 48, B and B r , and thus demagnetizes the field in a generator B, magnetizes it in a synchronous motor B f . If the armature current leads the nominal generated e.m.f., it reaches its maximum in an earlier position, while the arma- ture coil still partly faces the pre- ceding magnet pole, as shown in Fig. 48, C and C", and thus mag- netizes the field in a generator, Fig. 48, C, and demagnetizes it in a syn- chronous motor C'. With non-inductive load, or with the current in phase with the ter- minal voltage of an alternating- current generator, the current lags behind the nominal generated e.m.f., due to armature reaction and self- inductance, and thus partly de- magnetizes; that is, the voltage is lower under load than at no load with the same field excitation. In other words, lagging current demag- netizes and leading current magne- tizes the field of an alternating-cur- rent generator, while the opposite is the case with a synchronous motor. 10. In Fig. 49 let OF = F = resultant m.m.f. of field exci- tation and armature current (the m.m.f. of the field excita- tion being alternating with regard to the armature coil, due to its rotation) and 2 the lag of the current / behind the virtual e.m.f. E<2, generated by the resultant m.m.f. The virtual e.m.f. E 2 lags in time 90 degrees behind the result- ant flux of OF, and is thus represented by OE 2 in Fig. 47, and the m.m.f. of the armature current F a by OF a , lagging by angle 2 behind OE 2 . The resultant m.m.f. OF is the diagonal of the parallelogram with the component m.m.fs. OF a = armature m.m.f. and OF Q = total impressed m.m.f. or field excitation, as FIG. 49. Diagram of m.m.fs. in loaded generator. 132 ELEMENTS OF ELECTRICAL ENGINEERING sides, and from this construction OF Q is found. OF is thus the position of the field pole with regard to the armature. It is trigonometrically, If I = current per armature turn in amperes effective, n = number of turns per pole in a single-phase alternator, the arma- ture reaction is F a = nl ampere-turns effective, and is pulsating between zero and nl \/2. In a quarter-phase alternator with n turns per pole and phase in series and I amperes effective per turn, the armature reaction per phase is nl ampere-turns effective and nl \/2 ampere-turns maximum. The two phases magnetize in quad- rature, in phase and in space. Thus, at the time t, correspond- ing to angle 6 after the maximum of the first phase, the m.m.f. in the direction by angle 6 behind the direction of the magnetiza- tion of the first phase is nl \/2 cos 2 6. The m.m.f. of the second phase is nl \/2 sin 2 0; thus the total m.m.f. or the armature reaction F a = nl \/2, and is constant in intensity, but revolves synchronously with regard to the armature; that is, it is station- ary with regard to the field. In a three-phaser of n turns in series per pole and phase and / amperes effective per turn, the m.m.f. of each phase is nl \/2 ampere-turns maximum; thus at angle 6 in positon and angle in time behind the maximum of one phase; The m.m.f. of this phase is nl \/2 cos 2 0. The m.m.f. of the second phase is nl V2 cos 2 (0 + 120) = n/V2 ( - 0.5 cos - 0.5 \/3 sin 0)*. The m.m.f. of the third phase is nl V2 cos 2 (0 + 240) = nl \/2 ( - 0.5 cos + 0.5 \/3 sin 0) 2 . Thus the total m.m.f. or armature reaction, F a = nl \/2 (cos 2 + 0.25 cos 2 + 0.75 sin 2 + 0.25 cos 2 + 0.75 sin 2 0) = l.Snl \/2, constant in intensity, but revolving synchronously with regard to the armature, that is, stationary with regard to the field. These values of armature reaction correspond strictly only to the case where all conductors of the same phase are massed SYNCHRONOUS MACHINES 133 together in one slot. If the conductors of each phase are dis- tributed over a greater part of the armature surface, the values of armature reaction have to be multiplied by the average cosine of the total angle of spread of each phase. 11. The single-phase machine thus differs from the poly- phase machines: in the latter, on balanced load, the armature reaction is constant, while in the single-phase machine the armature reaction and thereby the resultant m.m.f. of field and armature is pulsating. The pulsation of the resultant m.m.f. of the single-phase machine causes a pulsation of its magnetic field under load, of double frequency, which generates a third harmonic of e.m.f. in the armature conductors. In machines of high armature reaction, as steam-turbine-driven single-phase alternators, the pulsation of the magnetic field may be sufficient to cause serious energy losses and heating by eddy currents, and thus has to be checked. This is usually done by a squirrel- cage induction machine winding in the field pole faces, or by short-circuited conductors laid in the pole faces in electrical space quadrature to the field coils. In these conductors, secondary currents Ei'_ of double frequency are produced which equalize the resultant m.m.f. of the machine. IV. Self-inductance 12. The effect of self -inductance is ^ similar to that of armature reaction, FlQ 50 ._ Diagram of e m fs> and depends upon the phase relation in loaded generator, in the same manner. If EI = real generated voltage, 0i = lag of current behind generated voltage EI, the magnetic flux produced by the arma- ture current I is in phase with the current, and thus the counter e.m.f. of self-inductance is in quadrature behind the current, and therefore the e.m.f. consumed by self-inductance is in quadrature ahead of the current. Thus in Fig. 50, denoting OEi = EI the generated e.m.f., the current is 01 = 7 ; lagging 61 behind OEi, the e.m.f. consumed by self -inductance OE "i, is 90 degrees ahead of the current, and the virtual generated e.m.f. E 2 , is the resultant of OEi and OE'\. As seen, the diagram of e.m.f s. of self -induc- tance is similar to the diagram of m.m.fs. of armature reaction. 134 ELEMENTS OF ELECTRICAL ENGINEERING 13. From this diagram we get the effect of load and phase re- lation npon the e.m.f. of an alternating-current generator. Let E terminal voltage per machine circuit, 7 = current per machine circuit, and = lag of the current behind the terminal voltage. Let r = resistance, x = reactance of the alternator armature. FIG. 51. Diagram showing combined effect of armature reaction and arma- ture self-inductance. Then, in the vector diagram, Fig. 51, OE = E, the terminal voltage, assumed as zero vector. 01 = I, the current, lagging by the angle EOI = 0. _The e.m.f. consumed by resistance is OE \ = Ir in phase with 01. The e-m-i^ consumed by reactance is OE f z Ix, 90 degrees ahead of 01. The real generated e.m.f. is found by combining OE and OE\ to SYNCHRONOUS MACHINES 135 The virtual generated e.m.f. is OEi and OE' Z combined to = E 2 . The m.m.f. required to produce -this e.m.f. Ez is OF = F, Fa I E, FIG. 52. Diagram of generator e.m.fs. and m.m.fs. for non-reactive load. 90 deg. ahead of OE 2 . It is the resultant of the armature m.m.f. or armature reaction and of the impressed m.m.f. or field excita- tion. The armature m.m.f. is in phase with the cur- rent 7, and is nl in a single-phase machine, n \/2 / in a quarter-phase machine, 1.5 \/2 nl in a three- phase machine, if n = number of armature turns per pole and phase. The m.m.f. of armature reaction is represented in the diagram by OF a of F a in phase with 01, and the impressed m.m.f. or field excitation OFo = FQ is the side of a parallelogram with OF as diag- onal and OF a as other side; or, the m.m.f. consumed by armature reaction is represented by OF' a = F a in opposition to 01. Combining OF' a and OF gives OF Q = FQ, the field excitation. F, FIG. 53. Diagram of generator, e.m.fs. and m.m.fs. for lagging reac- tive load. Power-factor . 50. FIG. 54. Diagram of generator e.m.fs. and m.m.fs. for leading reac- tive load. Power-factor 0.50. 136 ELEMENTS OF ELECTRICAL ENGINEERING In Figs. 52, 53, 54 are drawn the diagrams for = zero or non-inductive load, = 60 degrees, or 60 degrees lag (inductive load of power-factor 0.50), and = 60 deg., or 60 deg. lead (anti-inductive load of power-factor 0.50). Thus it is seen that with the same terminal voltage E a much higher field excitation, FQ, is required with inductive load than with non-inductive load, while with anti-inductive load a much lower field excitation is required. With open circuit the field excitation required to produce the terminal voltage W E would be ~r F = FQQ, or less than the field excitation J^o with JCJQ non-inductive load. Inversely, with constant field excitation, the voltage of an al- ternator drops with non-inductive load, drops much more with inductive load, and drops less, or even rises, with anti-inductive load. V. Synchronous Reactance 14. In general, both effects, armature self-inductance and armature reaction, can be combined by the term " synchronous reactance." FIG. 55. Diagram showing effect of synchronous reactance. FIG. 56. Diagram of generator e.m.f s. showing affect of synchronous reactance with non-reactive load. In a polyphase machine, the synchronous reactance is different, and lower, with one phase only loaded, as " single-phase synchro- nous reactance," than with all phases uniformly loaded, as " poly- phase synchronous reactance." The resultant armature reac- tion of all phases of the polyphase machine is higher than that with the same current in one phase only, and so also the self- SYNCHRONOUS MACHINES 137 inductive flux, as resultant flux of several phases, and thus rep- resents a higher synchronous reactance. Let r = effective resistance, XQ = synchronous reactance of armature, as discussed in Section II. Let E = terminal voltage, / = current, = angle of lag of the current behind ' the terminal vol- tage. It is in vector diagram, Fig. 55. OE = E = terminal voltage assumed as zero vector. 01 = FIG. 57. Diagram of generator e.m.fs. showing effect of synchronous reactance with lagging reactive load. 6 = 60 degrees. FIG. 58. Diagram of generator e.m.fs. Showing effect of synchro- nous reactance with leading reactive load 6 = 60 degrees. I = current lagging by the angle EOI = behind the terminal voltage. OE\ = Ir is the e.m.f. consumed by resistance, in phase with 01 j and OE'o = Ix the e.m.f. consumed by the synchronous reactance, 90 degrees ahead of the_current OI. OE'i and OE' Q combined give OE' = E' the e.m.f. consumed by the synchronous impedance. Combining OE'i, OE'o, OE gives the nominal generated e.m.f. OEo = EQ, corresponding to the field excitation F Q . In Figs. 56, 57, 58, are shown the diagrams for 6 = or non- inductive load, 6 = 60 degrees lag or inductive load, and & 60 degrees or anti-inductive load. Resolving all e.m.fs. into components in phase and in quad- rature with the current, or into power and reactive components, in symbolic expression we have: 138 ELEMENTS OF ELECTRICAL ENGINEERING the terminal voltage E = E cos 6 + jE sin 6 ; the e.m.f. consumed by resistance, E\ = ir; the e.m.f. consumed by synchronous reactance, E' = + jix Q , and the nominal generated e.m.f., E = E + E\ + E' Q = (E cos + ir) + j (E sin 19 + ix ) ; or, since . , , , / power current \ cos 6 = p = power-factor of the load ( = -. . ) \ total current / and q = \/l p 2 = sin = inductance factor of the load (wattless current\ total current' / ' it is Eo = (Ep + + j (Eq + ix ), or, in absolute values, Eo = V(Ep + ir) 2 + (Eq + ^ ) 2 ; hence, E = VE 2 - i 2 (x p - rq) 2 - i (rp -f x^q). The power delivered by the alternator into the external cir- cuit is P = iEp; that is, the current times the power component of the terminal voltage. The electric power produced in the alternator armature is Po = i(Ep + ir)-, that is, the current times the power component of the nomi- nal generated e.m.f., or, what is the same thing, the current times the power component of the real generated e.m.f. VI. Characteristic Curves of Alternating-current Generator 15. In Fig. 59 are shown, at constant terminal voltage E, the values of nominal generated e.m.f. E , and thus of field excitation FQ, with the current 7 as abscissas and for the three conditions, 1. Non-inductive load, p = 1, q = 0. 2. Inductive load of = 60 degrees lag, p = 0.5, q = 0.866. 3. Anti-inductive load of 6 = 60 degrees lead, p = 0.5, q = -0.866. SYNCHRONOUS MACHINES 139 The values r = 0.1, XQ = 5, E = 1000, are assumed. These curves are called the compounding curves of the synchronous generator. In Fig. 60 are shown, at constant nominal generated e.m.f. EQ, that is, at constant field excitation F , the values of terminal vol- E = 000 =5, '*& z EO.F, 500 20 10 00 80 100 120 UO 160 180 200AMP. FIG. 59. Synchronous generator compounding curves. tage E with the current I as abscissas and for the same resistance and synchronous reactance r = 0.1, XQ = 5, for the three different conditions, 1. Non-inductive load, p = 1, q = 0, E Q = 1127. 2. Inductive load of 60 degrees lag, p = Q.5, q = 0.866, E = 1458. 140 ELEMENTS OF ELECTRICAL ENGINEERING 3. Anti-inductive load of 60 degrees lead, p = 0.5, q = -0.866, E = 628. The values of E Q (and thus of F Q ) are assumed so as to give E = 1000 at I = 100. These curves are called the regulation curves of the alternator, or the field characteristics of the syn- chronous generator. In Fig. 61 are shown the load curves of the machine, with the 40 60 80 100 120 140 160 180 200 220 240 260 280 AMP. FIG. 60. Synchronous generator regulation curves. current I as abscissas and the watts output as ordinates corre- sponding to the same three conditions as Fig. 60. From the field characteristics of the alternator are derived the open-cir- cuit voltage of 1127 at full non-inductive load excitation, which is 1.127 times full-load voltage; the short-circuit current 225 at full non-inductive load excitation, which is 2.25 times full-load current; and the maximum output, 124 kw., at full non-induct- ive load excitation, which is 1.24 times rated output, at 775 volts and 160 amp. It depends upon the point on the field SYNCHRONOUS MACHINES 141 characteristic at which the alternator works, whether it tends to regulate for, that is, maintains, constant voltage, or constant current, or constant power, approximately. z L 7 \ H 20 40 60 80 100 120 140 160 180 200 220 240 260 280 AMP. FIG. 61. Synchronous generator load curves. VII. Synchronous Motor 16. As seen in the preceding, in an alternating-current gen- erator the field excitation required for a given terminal voltage and current depends upon the phase relation of the external circuit or the load. Inversely, in a synchronous motor the phase relation of the current into the armature at a given ter- minal voltage depends upon the field excitation and the load. Thus, if E = terminal voltage or impressed e.m.f., I = current, 6 = lag of current behind impressed e.m.f. in a synchronous motor of resistance r and synchronous reactance X Q , the polar diagram is as follows, Fig. 62. OE = E is the terminal voltage assumed as zero vector. The current 01 = I lags by the angle EOI = 6. The e.m.f. consumed by resistance isj9#'i = Ir. The e.m.f. consumed by synchronous reactance, OE'o = Ix Q . Thus, com- 142 ELEMENTS OF ELECTRICAL ENGINEERING bining OE'i and OE'o gives OE', the e.m.f. consumed by the synchronous impedance. The e.m.f. consumed by the synchro- nous impedance OE' and the e.m.f. consumed by the nominal generated or counter e.m.f. of the synchronous motor OEo, combined, give the impressed e.m.f. OE. Hence OEo is one side of a parallelogram, with OE' as the other side, and OE as diagonal. OEoo .(not shown), equal and opposite OE , would thus be the nominal counter-generated e.m.f. of the synchronous motor. In Figs. 63 to 65 are shown the polar diagrams of the syn- chronous motor for 6 = deg., 6 = 60 deg., 6 = 60 deg. It is seen that the field excitation has to be higher with lead- d E' FIG. 62. Vector diagram of synchronous motor. FIG. 63. Vector diagram of synchronous motor. 0=0 ing and lower with lagging current in a synchronous motor, while the opposite is the case in an alternating-current generator. In symbolic representation, by resolving all e.m.fs. into power components in phase with the current and wattless components in quadrature with the current i, we have: the terminal voltage, E = E cos 6 + jE sin 6 = Ep + jEq; the e.m.f. consumed by resistance, E/i = ir, and the e.m.f. consumed by synchronous reactance, E'Q = + jix . Thus the e.m.f. consumed by the nominal counter-generated e.m.f. is Eo = E - E'i - E' Q = (E cos - ir) + j (E sin 6 - ix Q ) = (Ep - ir) + j(Eq - ix Q ); SYNCHRONOUS MACHINES 143 or, in absolute values, V(Ecos e - ir) 2 + (Esin 6 - ix )* = V(Ep- ij hence, E = i (rp + x Q q) \/E Q 2 i 2 (x p rq) z . The power consumed by the synchronous motor is P = iEp; that is, the current times the power component of the impressed e.m.f. /" Eo FIG. 64. Vector diagram of syn- FIG. 65. Vector" diagram of synchronous chronous motor. 6 = 60 deg. motor. = 60 degrees. The mechanical power delivered by the synchronous motor armature is Po = i(Ep-ir); that is, the current times the power component of the nominal counter-generated e.m.f. Obviously to get the available mechan- ical power, the power consumed by mechanical friction and by molecular magnetic friction or hysteresis, and the power of field excitation, have to be subtracted from this value P . VIII. Characteristic Curves of Synchronous Motor 17. In Fig. 66 are shown, at constant impressed e.m.f. E, the nominal counter-generated e.m.f. E Q and thus the field excitation F Q required, 1. At no phase displacement, 6 = 0, or for the condition of minimum input; 144 ELEMENTS OF ELECTRICAL ENGINEERING 2. For = + 60, or 60 deg. lag: p = 0.5, q = + 0.866, and 3. For = - 60, or 60 deg. lead: p = 0.5, q = - 0.866, with the current I as abscissas, the constants being r = 0.1, z = 5, and E = 1000. These curves are called the compounding curves of the syn- chronous motors. In Fig. 67 are shown, with the power output PI = i (Ep ir) (iron loss and friction) as abscissas, and the same constants 1= E = =0.1, 000 X Q = 1100 20 40 60 80 100 120 140 160 180 200 FIG. 66. Synchronous motor compounding curves. r = 0.1, XQ = 5, E = 1000, and constant field excitation F ' that is, constant nominal counter-generated e.m.f. E Q = 1109 (corresponding to p = 1, # = at 7 = 100), the values of current I and power-factor p. As iron loss is assumed 3000 watts, as friction 2000 watts. Such curves are called load characteristics of the synchronous motor. 18. In Fig. 68 are shown, with constant power output = PO, SYNCHRONOUS MACHINES 145 i (Ep ir), and the same constants, r = 0.1, XQ = 5, E = 1000, and with the nominal counter-generated voltage E , that is, field excitation F Q , as abscissas, the values of current / for the four conditions, PO = 5 kw., or PI = 0, or no load, Po = 50 kw., or Pi = 45 kw., or half load, Po = 95 kw., or Pi = 90 kw., or full load, Po = 140 kw., or PI = 135 kw., or 150 per cent, of load. 10 20 30 40 50 60 70 80 90 100 110 120 130 140 FIG. 67. Synchronous motor load characteristics. Such curves are called phase characteristics of the synchronous motor. We have Po = iEp - i*r. Hence, Po + iV P = E Q = - p (Eq - ix ) 2 . Similar phase characteristics exist also for the synchronous generator, but are of less interest. It is seen that on each of the four-phase characteristics a certain field excitation gives 146 ELEMENTS OF ELECTRICAL ENGINEERING minimum current, a lesser excitation gives lagging current, a greater excitation leading current. The higher the synchronous reactance XQ, and thus the armature reaction of the synchronous motor, the flatter are the phase characteristics; that is, the less sensitive is the synchronous motor for a change of field excitation or of impressed e.m.f. Thus a relatively high armature reaction is desirable in a synchronous motor to secure stability, that is, independence of minor fluctuations of impressed voltage or of field excitation. 19. The theoretical maximum output of the synchronous motor, or the load at which it drops out of step, at constant impressed voltage and frequency is, even with very high armature reaction, usually far beyond the heating limits of the machine. 200 100 600 800 1000 1200 UOO 1600 1800 2000 FIG. 66. Synchronous motor phase characteristics. The actual maximum output depends on the drop of terminal voltage due to the increase of current, and on the steadiness or uniformity of the impressed frequency, thus upon the individual conditions of operation, but is as a rule far above full load. Hence, by varying the field excitation of the synchronous motor the current can be made leading or lagging at will, and the syn- chronous motor thus offers the simplest means of producing out of phase or wattless currents for controlling the voltage in trans- mission lines, compensating for wattless currents of induction motors, etc. Synchronous machines used merely for supplying wattless currents, that is, synchronous motors or generators running light, with over-excited or under-excited field, are called synchronous condensers. They are used as exciters for induc- tion generators, as compensators for the reactive lagging currents SYNCHRONOUS MACHINES 147 of induction motors, for voltage control of transmission lines, etc. Sometimes they are called "rotary condensers" or "dynamic condensers" when used only for producing lead- ing currents. IX. Magnetic Characteristic or Saturation Curve 20. The dependence of the generated e.m.f., or terminal voltage at open circuit, upon the field excitation is called the magnetic characteristic, or saturation curve, of the synchronous 1000 2000 3000 4000 5000 FIG. 69. Synchronous generator magnetic 7000 characteristics. machine. It has the same general shape as the curve of mag- netic flux density, consisting of a straight part below saturation, a bend or knee, and a saturated part beyond the knee. Gener- ally the change from the unsaturated to the over-saturated por- tion of the curve is more gradual; thus the knee is less pronounced in the magnetic characteristic of the synchronous machines, since the different parts of the magnetic circuit approach saturation successively. The dependence of the terminal voltage upon the field excita- tion, at constant full-load current through the amature into a 148 ELEMENTS OF ELECTRICAL ENGINEERING non-inductive circuit, is called the load saturation curve of the synchronous machine. It is a curve approximately parallel to the no-load saturation curve, but starting at a definite value of field excitation for zero terminal voltage, the field excitation required to maintain full-load current through the armature against its synchronous impedance. dF dE The ratio - -=- ~FT r Hi is called the saturation factor s of the machine. It gives the ratio of the proportional change of field excitation required for a change of voltage. The quantity 5 = 1 is called the percentage saturation of the machine, as it shows the approach to saturation. In Fig. 69 is shown the magnetic characteristic or no-load saturation curve of a synchronous generator, the load satura- tion curve and the no-load saturation factor, assuming E = 1000, I = 100 as full-load values. In the preceding the characteristic curves of synchronous ma- chines were discussed under the assumption that the saturation curve is a straight line ; that is, the synchronous machines working below saturation. 21. The effect of saturation on the characteristic curves of synchronous machines is as follows: The compounding curve is impaired by saturation; that is, a greater change of field excita- tion is required with changes of load. Under load the magnetic density in the armature corresponds to the true generated e.m.f. EI, the magnetic density in the field to the virtual generated e.m.f. EI. Both, especially the latter, are higher than the no- load e.m.f. or terminal voltage E of the generator, and thus a greater increase of field excitation is required in the presence of saturation than in the absence thereof. In addition thereto, due to the counter m.m.f. of the armature current, the magnetic stray field, that is, that magnetic flux which leaks from field pole to field pole through the air, increases under load, especially with inductive load where the armature m.m.f. directly opposes the field, and thus a still further increase of density is required in the field magnetic circuit under load. In consequence thereof, at high saturation the load saturation curve differs more from the no-load saturation curve than corresponds to the synchronous impedance of the machine. SYNCHRONOUS MACHINES 149 The regulation becomes better by saturation; that is, the increase of voltage from full load to no load at constant field excitation is reduced, the voltage being limited by saturation. Owing to the greater difference of field excitation between no load and full load in the case of magnetic saturation, the improvement in regulation is somewhat reduced. X. Efficiency and Losses 22. Besides the above described curves the efficiency curves are of interest. The efficiency of alternators and synchronous motors is usually so high that a direct determination by measuring the mechanical power and the electric power is less reliable than 10 20 30 4ff 50 60 TO 80 90 100 UO 120 130 140 150 160 170 ISO 150 200 KW. FIG. 70. Synchronous generator, efficiency and losses. the method of adding the losses, and the latter is therefore com- monly used. The losses consist of the following: the resistance loss in the armature; the resistance loss in the field circuit; the hysteresis and eddy current losses in the magnetic circuit; the friction and windage losses, and eventually load losses, that is, losses due to eddy currents and hysteresis produced by the load current in the armature. The resistance loss in the armature is proportional to the square of the current, I. The resistance loss in the field circuit is proportional to the square of the field excitation current, that is, the square of the nominal generated or counter-generated e.m.f., E Q . 10 150 ELEMENTS OF ELECTRICAL ENGINEERING The hysteresis loss is proportional to the 1.6th power of the real generated e.m.f., E l = E Ir. The eddy current loss is usually proportional to the square of the generated e.m.f., E\. The friction and windage loss is assumed as constant. The load losses vary more or less proportionally to the square of the current in the armature, and should be small with proper design. They can often be represented by an "effective" arma- ture resistance. Assuming in the preceding example a friction loss of 2000 watts; an iron loss of 3000 watts, at the generated e.m.f. EI = 1000; a 10 30 30 40 50 60 70 1.00 UO 120 130 140 150 100 170 ISO 190 200 K.W, FIG. 71. Synchronous motor efficiency and losses. resistance loss in the field circuit of 800 watts, at E Q = 1000, and a load loss at full load of 600 watts. The loss curves and efficiency curves are plotted in Fig. 70 for the generator, with the current output at non-inductive load or = as abscissas, and in Fig. 71 for the synchronous motor, with the mechanical power output as abscissas. XI. Unbalancing of Polyphase Synchronous Machines 23. The preceding discussion applies to polyphase as well as single-phase machines. In polyphase machines the nominal generated e.m.fs. or nominal counter-generated e.m.fs. are neces- sarily the same in all phases (or bear a constant relation to each other). Thus in a polyphase generator, if the current or the SYNCHRONOUS MACHINES 151 phase relation of the current is different in the different branches, the terminal voltage must become different also, more or less. This is called the unbalancing of the polyphase generator. It is due to different load or load of different inductance factor in the different branches. Inversely, in a polyphase synchronous motor, if the terminal voltages of the different branches are unequal, due to an unbal- ancing of the polyphase circuit, the synchronous motor takes more current or lagging current from the branch of higher vol- tage, and thereby reduces its voltage, and takes less current or leading current 1 from the branch of lower voltage, or even returns current into this branch, and thus raises its voltage. Hence a synchronous motor tends to restore the balance of an unbalanced polyphase system ; that is, it reduces the unbalancing of a polyphase circuit caused by an unequal distribution or unequal phase relation of the load on the different branches. To a less degree the induction motor possesses the same property. XII. Starting of Synchronous Motors 24. In starting, an essential difference exists between the single- phase and the polyphase synchronous motor, in so far as the for- mer is not self-starting but has to be brought to complete syn- chronism, or in step with the generator, by external means before it can develop torque, while the polyphase synchronous motor starts from rest and runs up to synchronism with more or less torque. In starting, the field excitation of the polyphase synchronous motor should be zero or very low. The starting torque is due to the magnetic attraction of the armature currents upon the remanent magnetism left in the field poles by the currents of the preceding phase, and to the eddy currents produced therein. Let Fig. 72 represent the magnetic circuit of a polyphase synchronous motor. The m.m.f. of the polyphase armature currents acting upon the successive projections or teeth of the armature, 1, 2, 3, etc., reaches a maximum in them successively; that is, the armature is the seat of a m.m.f. rotating synchro- nously in the direction of the arrow A. The magnetism in the 1 Since with lower impressed voltage the current is leading, with higher impressed voltage lagging, in a synchronous motor. 152 ELEMENTS OF ELECTRICAL ENGINEERING face of the field pole opposite to the armature projections lags behind the m.m.f., due to hysteresis and eddy currents, and thus is still remanent, while the m.m.f. of the projection 1 decreases, and is attracted by the rising m.m.f. of projection 2, etc., or, in other words, while the maximum m.m.f. in the armature has a position a, the maximum magnetism in the field-pole face still has the position b, and is thus attracted toward a, causing the field to revolve in the direction of the arrow A (or with a station- ary field, the armature to revolve in the opposite direction B). Lamination of the field poles reduces the starting torque caused by eddy currents in the field poles, but increases that caused by remanent magnetism or hysteresis, due to the higher permeability of the field poles. Thus the torque per volt-ampere input is approximately the same in either case, but with laminated i FIG. 72. Magnetic circuit of a polyphase synchronous motor. poles the impressed voltage required in starting is higher and the current lower than with solid field poles. In either case, at full impressed e.m.f. the starting current of a synchronous motor is large, since in the absence of a counter e.m.f. the total impressed e.m.f. has to be consumed by the impedance of the armature cir- cuit. Since the starting torque of the synchronous motor is due to the magnetic flux produced by the alternating armature cur- rents, or the armature reaction, synchronous motors of high armature reaction are superior in starting torque. Very frequently in synchronous motors a squirrel-cage wind- ing is used in the field pole faces, to give powerful starting torque by the induced currents therein, on the induction motor principle. Such squirrel-cage winding should have fairly high resistance to start well from rest, but low resistance to give powerful syn- chronizing, that is, to pull its load promptly into synchronism. SYNCHRONOUS MACHINES 153 XIII. Parallel Operation 25. Any alternator can be operated in parallel, or synchronized with any other alternator. A single-phase machine can be syn- chronized with one phase of a polyphase machine, or a quarter- phase machine operated in parallel with a three-phase machine by synchronizing one phase of the former with one phase of the latter. Since alternators in parallel must be in step with each other and have the same terminal voltage, the condition of satis- factory parallel operation is that the frequency of the machines is identically the same, and the field excitation such as would give the same terminal voltage. If this is not the case, there will be cross currents between the alternators in a local circuit; that is, the alternators are not without current at no load, and their currents under load are not of the same phase and proportional to their respective capacities. The cross currents between alternators when operated in parallel can be wattless currents or power currents. If the frequencies of two alternators are identically the same, but the field excitation not such as would give equal terminal voltage when operated in parallel, there is a local current between the two machines which is wattless and leading or magnetizing in the machine of lower field excitation, lagging or demagnetiz- ing in the machine of higher field excitation. At load this watt- less current is superimposed upon the currents from the machines into the external circuit. In consequence thereof the current in the machine of higher field excitation is lagging behind the cur- rent in the external circuit, the current in the machine of lower field excitation leads the current in the external circuit. The currents in the two machines are thus out of phase with each other, and their sum larger than the joint current, or current in the external circuit. Since it is the armature reaction of leading or lagging current which makes up the difference between the impressed field excitation and the field excitation required to give equal terminal voltage, it follows that the lower the armature reaction, that is, the closer the regulation of the machines, the more sensitive they are for inequalities or variations of field excitation. Thus, too low armature reaction is undesirable for parallel operation. With identical machines the changes in field excitation re- quired for changes of load must be the same. With machines 154 ELEMENTS OF ELECTRICAL ENGINEERING of different compounding curves the changes of field excitation for varying load must be different, and such as correspond to their respective compounding curves, if wattless currents shall be avoided. With machines of reasonable armature reaction the wattless cross currents are small even with relatively great inequality of field excitation. Machines of high armature re- action have been operated in parallel under circumstances where one machine was entirely without field excitation, while the other carried twice its normal field excitation, with wattless currents, however, of the same magnitude as full-load current. XIV. Division of Load in Parallel Operation 26. Much more important than equality of terminal voltage before synchronizing is equality of frequency. Inequality of frequency, or rather a tendency to inequality of frequency (since by necessity the machines hold each other in step or at equal frequency), causes cross currents which transfer'energy from the machine whose driving power tends to accelerate to the machine whose driving power tends to slow down, and thus relieves the latter by increasing the load on the former. Thus these cross currents are power currents, and cause at no load or light load the one machine to drive the other as synchronous motor, while under load the result is that the machines do not share the load in proportion to their respective capacities. The speed of the prime mover, as steam engine or turbine, changes with the load. The frequency of alternators driven thereby must be the same when in parallel. Thus their respect- ive loads are such as to give the same speed of the prime mover (or rather a speed corresponding to the same frequency) . Hence the division of load between alternators connected to independent prime movers depends almost exclusively upon the speed regula- tion of the prime movers. To make alternators divide the load in proportion to their capacities, the speed regulation of their prime movers must be the same; that is, the engines or turbines must drop in speed from no load to full load by the same percent- age and in the same manner. If the regulation of the prime movers is not the same, the load is not divided proportionally between the alternators, but the alternator connected to the prime mover of closer speed regula- tion takes more than its share of the load under heavy loads, and SYNCHRONOUS MACHINES 155 less under light loads. Thus, too close speed regulation of prime movers is not desirable in parallel operation of alternators. XV. Fluctuating Cross Currents in Parallel Operation 27. In alternators operated from independent prime movers, it is not sufficient that the average frequency corresponding to the average speed of the prime movers be the same, but still more important that the frequency be the same at any instant, that is, that the frequency (and thus the speed of the prime mover) be constant. In rotary prime movers, as turbines or electric motors, this is usually the case; but with reciprocating machines, as steam engines, the torque and thus the speed of rotation rises and falls periodically during each revolution, with the frequency of the engine impulses. The alternator con- nected with the engine will thus not have uniform frequency, but a frequency which pulsates, that is, rises and falls. The amplitude of this pulsation depends upon the design of the engine, the momentum of its fly-wheel, and the action of the engine governor. If two alternators directly connected to equal steam engines are synchronized so that the moments of maximum frequency coincide, there will be no energy cross currents between the machines, but the frequency of the whole system rises and falls periodically. In this case the engines are said to be synchronized. The parallel operation of the alternators is satisfactory in this case provided that the pulsations of engine speeds are of the same size and duration; but apparatus requiring constant fre- quency, as synchronous motors and rotary converters, when operated from such a system, will give a reduced maximum out- put, due to periodic cross currents between the generators of fluctuating frequency and the synchronous motors of constant frequency, and in an extreme case the voltage of the whole sys- tem will be caused to fluctuate periodically. Even with small fluctuations of engine speed the unsteadiness of current due to this source is noticeable in synchronous motors and synchronous converters. If the alternators happen to be synchronized in such a position that the moment of maximum speed of the one coincides with the moment of minimum speed of the other, alternately the one and then the other alternator will run ahead, and thus there 156 ELEMENTS OF ELECTRICAL ENGINEERING will be a pulsating power cross current between the alternators, transferring power from the leading to the lagging machine, that is, alternately from the one to the other, and inversely, with the frequency of the engine impulses. These pulsating cross currents are the most undesirable, since they tend to make the voltage fluctuate and to tear the alternators out of synchro- nism with each other, especially when the conditions are favorable to a cumulative increase of this effect by what may be called mechanical resonance (hunting) of the engine governors, etc. They depend upon the synchronous impedance of the alternators and upon their phase difference, that is, the number of poles and the fluctuation of speed, and are specially objectionable when operating synchronous apparatus in the system. 28. Thus, for instance, if two 80-pole alternators are directly connected to single-cylinder engines of 1 per cent, speed varia- tion per revolution, twice during each revolution the speed will rise, and fall twice; and consequently the speed of each alternator will be above average speed during a quarter revolution. Since the maximum speed is 1/2 per cent, above average, the mean speed during the quarter revolution of high speed is 1/4 per cent, above average speed, and by passing over 20 poles the armature of the machine will during this time run ahead of its mean posi- tion by 1/4 per cent, of 20 or 1/20 pole, that is, 180/20 = 9 elec- trical space degrees. If the armature of the other alternator at this moment is behind its average position by 9 electrical space degrees, the phase displacement between the alternator e.m.fs. is 18 electrical time degrees; that is, the alternator e.m.fs. are represented by OEi and OE Z in Fig. 71, and when running in parallel the e.m.f. OE f = E\E^ is short circuited through the synchronous impedance of the two alternators. . Since E' = OE\ = 2 EI sin 9 deg. the maximum cross current is ffisin9deg. 0.156 ffi 1 = = = U.loo 1 o, 20 20 ET where IQ = -- = short-circuit current of the alternator at full- 2 load excitation. Thus, if the short-circuit current of the alter- nator is only twice full-load current, the cross current is 31.2 per cent, of full-load current. If the short-circuit current is 6 times full-load current, the cross current is 93.6 per cent, of full-load current or practically equal to full-load current. Thus SYNCHRONOUS MACHINES 157 the smaller the armature reaction, or the better the regulation, the larger are the pulsating cross currents between the alternators, due to the inequality of the rate of rotation of the prime movers. Hence for satisfactory parallel operation of alternators connected to steam engines, a certain amount of armature reaction is de- sirable and very close regulation undesirable. By the transfer of energy between the machines the pulsations of frequency, and thus the cross currents, are reduced somewhat. Very high armature reaction is objectionable also, since it reduces the synchronizing power, that is, the tendency of the machines to hold each other in step, by reducing the energy transfer be- tween the machines. As seen herefrom, the problem of parallel operation of alternators is al- most entirely a problem of the regulation of their prime movers, especially steam A ^^ engines. With alternators driven by gas engines, the problem of parallel operation is made more difficult by the more jerky nature of the gas-engine ^ 73 ._ Phase displacement between impulse. In such machines, alternators to be synchronized, therefore, squirrel-cage wind- ings in the field-pole faces are commonly used, to assist synchron- izing by the currents induced in this short-circuited winding, on the principle of the induction machine. From Fig. 73 it is seen that the e.m.f. OE r or EiE 2 , which causes the cross current between two alternators in parallel con- nection, if their e.m.fs. OEi and OE% are out of phase, is approxi- mately in quadrature with the e.m.fs. OE\ and OE 2 of the machines, if these latter two e.m.fs. are equal to each other. The cross current between the machines lags behind the e.m.f. producing it, OE* ', by the angle co, where tan w = , and XQ = 7*0 reactance, r = effective resistance of alternator armature. The energy component of this cross current, or component in phase with OE f j is thus in quadrature with the machine voltages OEi and OE 2 , that is, transfers no power between them. The power transfer or equalization of load between the two machines takes place by the wattless or reactive component of cross current, E' 158 ELEMENTS OF ELECTRICAL ENGINEERING that is, the component which is in quadrature with OE', and thus in phase with one and in opposition with the other of the machine e.m.fs. OEi and OE^. 29. Hence, machines without reactance would have no syn- chronizing power, or could not be operated in parallel. The theoretical maximum synchronizing power exists if the reactance equals the resistance: XQ = r . This condition, however, cannot be realized, and if realized would give a dangerously high syn- chronizing power and cross current. In practice, XQ is always very much greater than ro, and the cross current thus practically in quadrature with OE', that is, in phase (or opposition) with the machine voltages, and is consequently an energy-transfer current. If, however, alternators are operated in parallel over a circuit of appreciable resistance, as two stations at some distance from each other are synchronized, especially if the resistance between the stations is non-inductive, as underground cables, with alter- nators of very low reactance, as turbo alternators, the synchro- nizing power may be insufficient. In this case, reactance has to be inserted between the stations, to lag the cross current and thereby make it a power-transferring or synchronizing current. If, however, the machine voltages OEi and OE Z are different in value but approximately in phase with each other, the voltage causing cross currents, E\Ez , is in phase with the machine vol- tages and the crosscurrents thus in quadrature with the machine voltages OEi and OE%, and hence do not transfer energy, but are wattless. In one machine the cross current is a lagging or demagnetizing, and in the other a leading or magnetizing, current. Hence two kinds of cross currents may exist in parallel opera- tion of alternators currents transferring power between the machines, due to phase displacement between their e.m.fs., and wattless currents transferring magnetization between the ma- chines, due to a difference of their induced e.m.fs. In compound-wound alternators, that is, alternators in which the field excitation is increased with the load by means of a series field excited by the rectified alternating current, it is al- most, but not quite, as necessary as in direct-current machines, when operating in parallel, to connect all the series fields in paral- lel by equalizers of negligible resistance, for the same reason to insure proper division of current between machines. SYNCHRONOUS MACHINES 159 XVI. Higher Frequency Cross Currents between Synchronous Machines 30. If several synchronous machines of different wave shapes are connected into the same circuit, cross currents exist between the machines of frequencies which are odd multiples of the circuit frequency, that is, higher harmonics thereof. The machines may be two or more generators, in the same or in different stations, of wave shapes containing higher harmonics of different order, intensity or phase, or synchronous motors or converters of wave shapes different from that of the system to which they are connected. The intensity of these cross currents is the difference of the corresponding harmonics of the machines divided by the impe- dance between the machines. This impedance includes the self- inductive reactance of the machine armatures. The reactance obviously is that at the frequency of the harmonic, that is, if x = reactance at fundamental frequency, it is nx for the nth harmonic. In most cases these cross currents are very small and negli- gible. With machines of distributed armature winding, the in- tensity of the harmonic is low, that is, the voltage nearly a sine wave, and with machines of massed armature winding, as uni- tooth alternators, the reactance is high. These cross currents thus usually are noticeable only at no load, and when adjusting the field excitation of the machines for minimum current. Thus in a synchronous motor or converter, at no load, the minimum current, reached by adjusting the field, while small compared with full-load current, may be several times larger than the minimum point of the " V" curve in Fig. 68, that is, the value of the energy current supplying the losses in the machine. It is only in the parallel operation of very large high-speed machines (steam turbine driven alternators) of high armature reaction and very low armature self-induction that such high- frequency cross currents may require consideration, and even then only in three-phase F-connected generators with grounded neutral, as cross currents between the neutrals of the machines. In a three-phase machine, the voltage between the terminals, or delta voltage, contains no third harmonic or its multiple, as the third harmonics of the Y voltage neutralize in the delta voltage, and such a machine, with a terminal voltage of almost sine shape, 160 ELEMENTS OF ELECTRICAL ENGINEERING may contain a considerable third harmonic in the Y voltage. As the three Y voltages of the three-phase system are 120 degrees apart in phase, their third harmonics are 3 X 120 deg. = 360 deg. apart, or in phase with each other, from the main terminals to the neutral, and by connecting the neutrals of two three- phase machines of different third harmonics with each other, as by grounding the neutrals, a cross current flows between the machines over the neutral, which may reach very high values. Even in machines of the same wave shape, such a triple frequency current appears between the machines over the neutral, when by a difference in field excitation a difference in the phase of the third harmonic is produced. It therefore is often undesirable to ground or connect together, without any resistance, the neutrals of three-phase machines, but in systems of grounded neutral either the neutral should be grounded through separate resist- ances or grounded only in one machine. XVII. Short-circuit Currents of Alternators 31. The short-circuit current of an alternator at full-load excitation usually is from two to five times full-load current, and even less in very large high-speed steam turbine alternators. It is where EQ = nominal generated e.m.f., ZQ = synchronous impe- dance of alternator, representing the combined effect of arma- ture reaction and armature self-inductance. In the first moment after short circuiting, however, the current frequently is many times larger than the permanent short- circuit current, that is, where z = self-inductive impedance of the alternator. That is, in the first moment after short circuiting the poly- phase alternator the armature current is limited only by the arma- ture self-inductance, and not by the armature reaction, and some appreciable time occasionally several seconds elapses before the armature reaction becomes effective. At short circuit, the magnetic field flux is greatly reduced by the demagnetizing action of the armature current, and the gen- SYNCHRONOUS MACHINES 161 erated e.m.f. thereby reduced from the nominal value E Q to the virtual value E z ; the latter is consumed by the armature self- inductive impedance z, or self-inductive reactance, which is practically the same in most cases. The armature self-inductance is instantaneous, since the magnetic field rises simultaneously with the armature current which produces it; armature reaction, however, requires an appre- ciable time to reduce the magnetic flux from the open-circuit value to the much lower short-circuit value, since the magnetic field flux is surrounded by the field exciting coils, which act as a short-circuited secondary opposing a rapid change of field flux; that is, in the moment when the short-circuit current starts it begins to demagnetize the field, and the magnetic field flux there- fore begins to decrease; in decreasing, however, it generates an e.m.f. in the field coils, which opposes the change of field flux, that is, increases the field current so as momentarily to main- tain the full field flux against the demagnetizing action of the armature reaction. In the first moment the armature current thus rises to the value given by the e.m.f. generated by the full field flux, while the field current rises, frequently to many times its normal value (hence, if circuit breakers are in the field circuit, they may open the circuit). Gradually the field flux decreases, and with it decrease the field current and the armature current to their normal values, at a rate depending on the resistance and the inductance of the field-exciting circuit. The decrease in value of the field flux will be the more rapid the higher the re- sistance of the field circuit, the slower the higher the inductance, that is, the greater the magnetic flux of the machine. Thus, the momentary short-circuit current of the machine can be made to decrease somewhat more rapidly by increasing the resistance of the field circuit, that is, wasting exciting power in the field rheostat. In the very first moment the short-circuit current waves are unsymmetrical, as they must simultaneously start from zero in all phases and gradually approach their symmetrical appear- ance, i.e., in a three-phase machine three currents displaced by 120 degrees. Hereby the field current is made pulsating, with nor- mal or synchronous frequency, that is, with the same frequency as the armature current. This full frequency pulsation gradually dies out and the field current becomes constant with a polyphase short circuit, while with a single-phase short circuit it remains 162 ELEMENTS OF ELECTRICAL ENGINEERING pulsating with double frequency, due to the pulsating armature reaction. In a polyphase short circuit this full frequency pul- sation due to the unsymmetrical starting of the currents is inde- pendent of the point of the wave at which the short circuit starts, since the resultant asymmetry of all the polyphase cur- rents is the same regardless of the point of the wave at which the circuit is closed. In a single-phase short circuit, however, the full frequency pulsation depends on the point of the wave at which the circuit is closed, and is absent if the circuit is closed at that moment at which the short-circuit current would pass through zero. The momentary short-circuit current of an alternator thus represents one of the few cases in which armature self-induc- tance and armature reaction do not act in the same manner, and the synchronous reactance can be split into two components, thus, XQ = x -\- x', where x = self-inductive reactance, which is due to a true self-inductance, and x' = effective reactance of armature reaction, which is not instantaneous. 32. In machines of high self-inductance and low armature re- action, as high frequency alternators, this momentary increase of short-circuit current over its normal value is negligible, and moderate in machines in which armature reaction and self-in- ductance are of the same magnitude, as large modern multi- polar low-speed alternators. In large high-speed alternators of high armature reaction and low self -inductance, as steam turbine alternators, the momentary short-circuit current may exceed the permanent value ten or more times. With such large currents magnetic saturation of the self-inductive armature circuit still further reduces the reactance x, that is, increases the current, and in such cases the mechanical shock on the generator becomes so enormous that it is necessary to reduce the momentary short-cir- cuit current by inserting self-inductance, that is, reactance coils into the generator leads, or by specifically designing the alterna- tor for high armature reactance, or by both. In view of the excessive momentary short-circuit current, it may be*desirable that automatic circuit breakers on such systems have a time limit, so as to keep the circuit closed until the short- circuit current has somewhat decreased. 33. In single-phase machines, and in polyphase machines in case of a short circuit on one phase only, the armature reaction is pulsating, and the field current in the first moment after the SYNCHRONOUS MACHINES 163 short circuit therefore pulsates, with double frequency, and remains pulsating even after the permanent condition has been reached. The double frequency pulsation of the field current in case of a single-phase short circuit generates in the armature a third harmonic of e.m.f. The short-circuit current wave be- comes greatly distorted thereby, showing the saw-tooth shape characteristics of the third harmonic, and in a polyphase machine on single-phase short circuit, in the phase in quadrature with the short-circuited phase, a very high voltage appears, which is greatly Field Current Armature Current FIG. 74. Three-phase short-circuit current in a turbo-alternator. distorted by the third harmonic and may reach several times the value of the open-circuit voltage. Thus, with a single- phase short circuit on a polyphase system, destructive voltages may appear in the open-circuited phase, of saw-tooth wave shape. Upon- this double frequency pulsation of the field current during a single-phase short circuit the transient full frequency pulsation resulting from the unsymmetrical start of the armature current is superimposed and thus causes a difference in the in- tensity of successive waves of the double frequency pulsation, 164 ELEMENTS OF ELECTRICAL ENGINEERING which gradually disappears with the dying out of the transient full frequency pulsation, and depends upon the point of the wave at which the short circuit is closed, and thus is absent, and the Armature current. Field Current FIG. 75. Single-phase short-circuit current in a three-phase turbo- alternator. .Armature current Field current 52.5 amp. EJ amp. FIG. 76. Single-phase short-circuit current in a three-phase turbo- alternator. double frequency pulsation symmetrical, if the circuit is closed at the moment when the short-circuit current should be zero. 34. As illustration is shown, in Fig. 74, the oscillogram of SYNCHRONOUS MACHINES 165 one phase of the three-phase short circuit of a three-phase turbo- alternator, giving the unsymmetrical start of the armature currents and the full frequency pulsation of the field current. In Fig. 75 is shown a single-phase short circuit of the same machine, in which the circuit is closed at the zero value of the current; the current wave therefore is symmetrical, and the field current shows only the double frequency pulsation due to the single-phase armature reaction. In Fig. 76 is shown another single-phase short circuit, in which the armature current wave starts unsymmetrical, thus giving a transient full frequency term in the field current. Thus in the double frequency pulsation of the field current at first large and small waves alternate, but the successive waves gradually be- come equal with the dying out of the full frequency term. In Figs. 75 and 76 the oscillogram is cut off by the open- ing of the circuit breaker. For further discussion, and the theoretical investigation of momentary short-circuit currents, see "Theory and Calculation of Transient Electric Phenomena and Oscillations," Part I, Chapters XI and XII. For further discussion of the terms reactance, armature re- action and field excitation and their relation, see "Theory and Calculation of Electric Circuits. " 11 B. DIRECT-CURRENT COMMUTATING MACHINES I. General 35. Commutating machines are characterized by the combina- tion of a continuously excited magnet field with a closed-circuit armature connected to a segmental commutator. According to their use, they can be divided into direct-current generators which transform mechanical power into electric power, direct- current motors which transform electric power into mechanical power, and direct-current con- verters which transform electric power into a different form of electric power. Since the most important class of the latter are the synchronous converters, which combine features of the synchronous machines with those of the commutating machines, they shall be treated in a sepa- rate chapter. By the excitation of their mag- net fields, commutating machines are divided into magneto machines, in which the field consists of permanent magnets; separately excited machines; shunt machines, in which the field is excited by an electric circuit shunted across the machine terminals, and thus receives a small branch current at full machine voltage, as shown diagrammatically in Fig. 77; series machines, in which the electric field circuit is connected in series with the armature, and thus receives the full machine cur- rent at low voltage (Fig. 78) ; and compound machines, excited by a combination of shunt and series field (Fig. 79). . In compound machines the two windings can magnetize either in the same direc- tion (cumulative compounding) or in opposite directions (dif- ferential compounding). Differential compounding has been used for constant-speed motors. Magneto machines are used only for very small sizes. 166 FIG. 77. Shunt machine. D. C. COMMUTATING MACHINES 167 36. By the number of poles commutating machines are divided into bipolar and multipolar machines. Bipolar machines are mainly used in small sizes. By the construction of the armature, commutating machines are divided into smooth-core machines and iron-clad or "toothed" armature machines. In the smooth- core machine the armature winding is arranged on the surface of a laminated iron core. In the iron-clad machine the arma- ture winding is sunk into slots. The iron-clad type has the ad- vantage of greater mechanical strength, but the disadvantage of higher self-inductance in commutation, and thus requires high- resistance, carbon or graphite, commutator brushes. The iron- clad type has the advantage of lesser magnetic stray field, due FIG. 78. Series machine. FIG. 79. Compound machine. to the shorter gap between field pole and armature iron, and of lesser magnet distortion under load, and thus can with carbon brushes be operated with constant position of brushes at all loads. In consequence thereof, for large multipolar machines the iron- clad type of armature is best adapted; the smooth-core type is hardly ever used nowadays. Either of these types can be drum wound or ring wound. The drum winding has the advantage of lesser self-inductance and lesser distortion of the magnetic field, and is generally less difficult to construct and thus mostly preferred. By the arma- ture winding, commutating machines are divided into multiple- wound and series-wound machines. The difference between multiple and series armature winding, and their modifications, can best be shown by diagram. 168 ELEMENTS OF ELECTRICAL ENGINEERING II. Armature Winding 37. Fig. 80 shows a six-pole multiple ring winding, and Fig. 81 a six-polar multiple drum winding. As seen, the armature coils are connected progressively all around the armature in closed circuit, and the connections between adjacent armature coils lead to the commutator. Such an armature winding has as many circuits in multiple, and requires as many sets of com- mutator brushes, as poles. Thirty-six coils are shown in Figs. 80 and 81, connected to 36 commutator segments, and the two sides of each coil distinguished by drawn and dotted lines. In a drum-wound machine, usually the one side of all coils forms the upper and the other side the lower layer of the armature winding. Fig. 82 shows a six-pole series drum winding with 36 slots and 36 commutator segments. In the series winding the circuit passes from one armature coil, not to the next adjacent armature coil as in the multiple winding, but first through all the armature coils having the same relative position with regard to the magnet poles of the same polarity, and then to the armature coil next ad- jacent to the first coil. That is, all armature coils having the same or approximately the same relative position to poles of equal polarity form one set of integral coils. Thus the series winding has only two circuits in multiple, and requires two sets of brushes only, but can be operated also with as many sets of brushes as poles, or any intermediate number of sets of brushes. In Fig. 82, a series winding in which the number of armature coils is divisible by the number of poles, the commutator segments have to be cross connected. Therefore this form of series winding is hardly ever used. The usual form of series winding is the winding shown by Fig. 83. This figure shows a six-polar armature having 35 coils and 35 commutator segments. In consequence thereof the armature coils under corresponding poles which are connected in series are slightly displaced from each other, so that after pass- ing around all corresponding poles the winding leads symmetric- ally into the coil adjacent to the first armature coil. Hereby the necessity of commutator cross connections is avoided, and the winding is perfectly symmetrical. With this form of series winding, which is mostly used, the number of armature coils must be chosen to follow certain rules. Generally the number of coils is one less or one more than a multiple of the number of poles. /). C. COMMUTATING MACHINES s 169 FIG. 80. Multiple ring armature winding. FIG. 81. Multiple drum full pitch winding. 170 ELEMENTS OF ELECTRICAL ENGINEERING FIG. 82. Series drum winding with cross-connected commutator. FIG. 83. Series drum winding. D. C. COMMUTATING MACHINES 171 All these windings are closed-circuit windings; that is, starting at any point, and following the armature conductor, the circuit returns into itself after passing all e.m.fs. twice in opposite direc- tion (thereby avoiding short circuit). An instance of an open- coil winding is shown in Fig. 84, a series-connected three-phase star winding similar to that used in the Thomson-Houston arc machine. Such open-coil windings, however, cannot be used in commutating machines. They are generally preferred in syn- chronous and in induction machines. FIG. 84. Open-circuit three-phase series drum winding. 38. By leaving space between adjacent coils of these windings a second winding can be laid in between. The second winding can either be entirely independent from the first winding, that is, each of the two windings closed upon itself, or after passing through the first winding the circuit enters the second winding, and after passing through the second winding it reenters the first winding. In the first case the winding is called a double spiral winding (or multiple spiral winding if more than two windings are used), in the latter case a double reentrant winding (or 172 ELEMENTS OF ELECTRICAL ENGINEERING multiple reentrant winding). In the double spiral winding the number of coils must be even; in the double reentrant winding, odd. Multiple spiral and multiple reentrant windings can be either multiple or series wound; that is, each spiral can consist either of a multiple or of a series winding. Fig. 85 shows a double spiral multiple ring winding, Fig. 86 a double spiral multiple drum winding, Fig. 87 a double reentrant multiple drum winding. As seen in the double spiral or double reentrant multiple wind- ing, twice as many circuits as poles are in multiple. Thus such FIG. 85. Multiple double spiral ring winding. windings are mostly used for large low-voltage machines, but as very few large direct-current generators are built nowadays, and alternating-current generation with synchronous converters usu- ally preferred, and as multiple spiral or reentrant windings are inconvenient in synchronous converters, their use has greatly decreased. 39. A distinction is frequently made between lap winding and wave winding. These are, however, not different types; but the wave winding is merely a constructive modification of the series drum winding with single-turn coil, as seen by comparing D. C. COMMUTATING MACHINES 173 FIG. 86. Multiple double spiral full pitch winding. FIG. 87. Multiple double re-entrant drum full pitch winding. 174 ELEMENTS OF ELECTRICAL ENGINEERING Figs. 88 and 89. Fig. 88 shows a part of a series drum winding developed. Coils C\ and C 2 , having corresponding positions under poles of equal polarity, are joined in series. Thus the end con- nection ah of coil Ci connects by cross connection be and cd to the FIG. 88. Series lap winding. end connection de of coil C%. If the armature coils consist of a single turn only, as in Fig. 86, and thus are open at 6 and d } the end connection and the cross connection can be combined by passing from a in coil Ci directly to c and from c directly to e in FIG. 89. Wave winding. coil C 2 ; that is, the circuit abcde is replaced by ace. This has the effect that the coils are apparently open at one side. Such a winding has been called a wave winding. Only series windings with a single turn per coil can be arranged as wave windings, while windings with several turns per coil must neces- D. C. COMMUTATING MACHINES 175 sarily be lap or coil windings. In Fig. 90 is shown a series drum winding with 35 coils and commutator segments, and a single turn per coil arranged as wave winding. This winding may be compared with the 35-coil series drum winding in Fig. 83. 40. Drum winding can be divided into full-pitch and frac- tional-pitch windings. In the full-pitch winding the spread of the coil covers the pitch of one pole; that is, each coil covers FIG. 90. Series drum wave winding. one-sixth of the armature circumference in a six-pole machine, etc. In a fractional-pitch winding it covers less or more. Series drum windings without cross-connected commutator in which thus the number of coils is not divisible by the number of poles are necessarily always slightly fractional pitch; but gen- erally the expression " fractional-pitch winding" is used only for windings in which the coil covers one or several teeth less than correspond to the pole pitch. Thus the multiple drum winding in Fig. 81 would be a fractional-pitch winding if the coils spread 176 ELEMENTS OF ELECTRICAL ENGINEERING over only four or five teeth instead of over six. As five-sixths fractional-pitch multiple drum winding it is shown in Fig. 91. Fractional-pitch windings have the advantage of shorter end connections and less self-inductance in commutation, since commutation of corresponding coils under different poles does not take place in the same, but in different, slots, and the flux of self-inductance in commutation is thus more subdivided. Fig. 91 shows the multiple drum winding of Fig. 81 as a frac- FIG. 91. Multiple drum five-sixth fractional pitch winding. tional-pitch winding with five teeth spread, or five-sixths pitch. During commutation the coils a b c d e f commutate simultane- ously. In Fig. 81 these coils lie by twos in the same slots, in Fig. 91 they lie in separate slots. Thus, in the former case the flux of self-inductance interlinked with the commutated coil is due to two coils; that is, twice that in the latter case. Frac- tional-pitch windings, however, have the disadvantage of reduc- ing the width of the neutral zone, or zone without generated e.m.f. between the poles, in which commutation takes place, D. C. COMMUTATING MACHINES 177 since the one side of the coil enters or leaves the field before the other. Therefore, in commutating machines it is seldom that a pitch is used that falls short of full pitch by more than one or two teeth, while in induction and synchronous machines occasionally as low a pitch as 50 per cent, is used, and two-thirds pitch is frequently employed. For special purposes, as in single-phase commutator motors fractional-pitch windings are sometimes used. 41. Series windings find their foremost application in machines with small currents, or small machines in which it is desirable to have as few circuits as possible in multiple, and in machines in which it is desirable to use only two sets of brushes, as in smaller railway motors. In multipolar machines with many sets of brushes a series winding is liable to give selective commutation; that is, the current does not divide evenly between the sets of brushes of equal polarity. Multiple windings are used for machines of large currents, thus generally for large machines, and in large low-voltage machines the still greater subdivision of circuits afforded by the multiple- spiral and the multiple-reentrant winding is resorted to. To resume, then, armature windings can be subdivided into (a) Ring and drum windings. (6) Closed-circuit and open-circuit windings. Only the former can be used for commutating machines. (c) Multiple and series windings. (d) Single-spiral, multiple-spiral, and multiple-reentrant wind- ings. Either of these can be multiple or series windings. (e) Full-pitch and fractional-pitch windings. III. Generated E.M.FS. 42. The formula for the generation of e.m.f. in a direct- current machine, as discussed in the preceding, is e = where e = generated e.m.f., / = frequency = number of pairs of poles X hundreds of rev. per sec., n = number of turns in series between brushes, and < = magnetic flux passing through the armature per pole, in megalines. In ring-wound machines, is one-half the flux per field pole, since the flux divides in the armature into two circuits, and each 178 ELEMENTS OF ELECTRICAL ENGINEERING armature turn incloses only half the flux per field pole. In ring- wound armatures, however, each armature turn has only one con- ductor lying on the armature surface, or face conductor, while in a drum-wound machine each turn has two face conductors. Thus, with the same . number of face conductors that is, the same armature surface the same frequency, and the same flux per field pole, the same e.m.f. is generated in the ring-wound as in the drum-wound armature. The number of turns in series between brushes, n, is one-half the total number of armature turns in a series-wound armature, - the total number of armature turns in a single-spiral multiple- wound armature with p poles. It is one-half as many in a double- spiral or double-reentrant, one-third as many in a triple-spiral winding, etc. By this formula, from frequency, series turns, and magnetic flux the e.m.f. is found, or inversely, from generated e.m.f., fre- quency, and series turns the magnetic flux per field pole is calculated: *--!-, 4/n From magnetic flux, and section and lengths of the different parts of the rnagnetic circuit, the densities and the ampere- turns required to produce these densities are derived, and as the sum of the ampere-turns required by the different parts of the magnetic circuit, the total ampere-turns excitation per field pole is found, which is required for generating the desired e.m.f. Since the formula for the generation of direct-current e.m.f is independent of the distribution of the magnetic flux, or its wave shape, the total magnetic flux, and thus the ampere-turns re- quired therefor, are independent also of the distribution of magnetic flux at the armature surface. The latter is of impor- tance, however, regarding armature reaction and commutation. IV. Distribution of Magnetic Flux 43. The distribution of magnetic flux in the air gap or at the armature surface can be calculated approximately by assuming the density at any point of the armature surface as proportional to the m.m.f. acting thereon, and inversely proportional to the nearest distance from a field pole. Thus, if F Q = ampere-turns D. C. COMMUTATING MACHINES 179 acting upon the air gap between armature and field pole, l a = length of air gap, from iron to iron, the density under the magnet pole, that is, in the range BC of Fig. 90, is At a point having the distance l x from the end of the field pole on the armature surface, the distance from the next field pole is l d = Vk 2 + l x 2 , and the density thus, approximately, B C FIG. 92. Distribution of mganetic flux under a single pole. Herefrom the distribution of magnetic flux is calculated and plotted in Fig. 92, for a single pole BC, along the armature sur- face A, for the length of air gap l a = 1, and such a m.m.f. as to L FIG. 93. Distribution of magnetic force and flux at no load. give Bo = 8000 under the field pole; that is, for / = 6400 or H Q = 8000. Around the surface of the direct-current machine armature, alternate poles follow each other. Thus the m.m.f. is constant only under each field pole, but decreases in the space between the field poles, from C to E in Fig. 93, from full value at C to full value in opposite direction at E. The point D midway 180 ELEMENTS OF ELECTRICAL ENGINEERING between C and E } at which the m.m.f. of the field equals zero, is called the "neutral." The distribution of m.m.f. of field excitation is thus given by the line F in Fig. 91. The distribu- tion of magnetic flux as shown in Fig. 91 by BQ is derived by the formula 4irF B 10 I where This distribution of magnetic flux applies only to the no-load condition. Under load, that is, if the armature carries current, the distribution of flux is changed by the m.m.f. of the armature current, or armature reaction. J FIG. 94. Distribution of flux with current in the armature. 44. Assuming the brushes set at the middle points between adjacent poles, D and G, Fig. 94, the m.m.f. of the armature is maximum at the point connected with the commutator brushes, in this case at the points D and G } and gradually decreases from full value at D to equal but opposite value at G, as shown by the line -F a in Fig. 94, while the line F Q gives the field m.m.f. or impressed m.m.f. If n = number of turns in series between brushes per pole, i = current per turn, the armature reaction is F a = ni ampere- turns. Adding F a and F gives the resultant m.m.f. F, and there- from the magnetic distribution: B = The latter is shown as line 10 l d in Fig. 94. D. C. COMMUTATING MACHINES 181 With the brushes set midway between adjacent field poles, the armature m.m.f. is additive on one side and subtractive on the other side of the center of the field pole. Thus the magnetic intensity is increased on one side and decreased on the other. The total m.m.f., however, and thus, neglecting saturation, the total flux entering the armature, are not changed. Thus, arma- ture reaction, with the brushes midway between adjacent field poles, acts distorting upon the field, but neither magnetizes nor demagnetizes, if the field is below saturation. The distortion of the magnetic field takes place by the arma- ture ampere-turns beneath the pole, or from B to C. Thus, if T = pole arc, that is, the angle covered by pole face (two poles or one complete period being denoted by 360 degrees), the dis- rF a torting ampere-turns of the armature reaction are As seen, in the assumed instance, Fig. 94, where F the m.m.f. at the two opposite pole corners, and thus the mag- netic densities, stand in the proportion 1 to 3. As seen, the generated e.m.f. is not changed by the armature reaction, with the brushes set midway between the field poles, except by the sftiall amount corresponding to the flux entering beyond D and G, that is, shifted beyond the position of brushes. At D, how- ever, the flux still enters the armature, depending in intensity upon the armature reaction; and thus with considerable arma- ture reaction the brushes when set at this point are liable to spark by short-circuiting an active e.m.f. Therefore, under load, the brushes are shifted toward the following pole, that is, toward the direction in which the zero point of magnetic flux has been shifted by the armature reaction. 45. In Fig. 95, the brushes are assumed as shifted to the cor- ner of the next pole, E respectively B. In consequence thereof, the subtractive range of the armature m.m.f. is larger than the additive, and the resultant m.m.f. F = F + F a is decreased; that is, with shifted brushes the armature reaction demagnet- izes the field. The demagnetizing armature ampere-turns are f~ir> PM = 7^TfF a . That is, if TI = angle of shift of brushes or angle of lead ( = GB in Fig. 95), assuming the pitch of two poles = 360 degrees, the demagnetizing component of armature reaction is 2 T V V *"; the distorting component is ^-^., where T = pole arc. loU loU 12 182 ELEMENTS OF ELECTRICAL ENGINEERING Thus, with shifted brushes the field excitation has to be in- creased under load to maintain the same total resultant m.m.f., that is, the same total flux and generated e.m.f. Hence, in *? jf ff Fig. 95 the field excitation F has been assumed by ** a = - loU o larger than in the previous figures, and the magnetic distribution BI plotted for these values. J FIG. 95. Distribution of flux with current in the armature and brushes shifted from the magnetic neutral. V. Effect of Saturation on Magnetic Distribution 46. The preceding discussion of Figs. 92 to 95 omits the effect of saturation. That is, the assumption is made that the mag- netic materials near the air gap, as pole face and armature teeth, are so far below saturation that at the demagnetized pole corner the magnetic density decreases, at the strengthened pole corner increases, proportionally to the m.m.f. The distribution of m.m.f. obviously is not affected by satu- ration, but the distribution of magnetic flux is greatly changed thereby. To investigate the effect of saturation, in Figs. 96 to 99 the assumption has been made that the air gap is reduced to one-half its previous value, l a = 0.5, thus consuming only one- half as many ampere-turns, and the other half of the ampere- turns are consumed by saturation of the armature teeth. The length of armature teeth is assumed as 3.2, and the space filled by the teeth is assumed as consisting of one-third of iron and two-thirds of non-magnetic material (armature slots, ventilating ducts, insulation between laminations, etc.). D. C. COMMUTATING MACHINES 183 In Figs. 96, 97, 98, 99, curves are plotted corresponding to those in Figs. 92, 93, 94, and 95. As seen, the spread of mag- netic flux at the pole corners is greatly increased, but farther away from the field poles the magnetic distribution is not changed. 47. The magnetizing, or rather demagnetizing, effect of the load with shifted brushes is not changed. The distorting effect FIG. 96. Flux distribution under a single pole. of the load is, however, very greatly decreased, to a small per- centage of its previous value, and the magnetic field under the field pole is very nearly uniform under load. The reason is: Even a very large increase of m.m.f. does not much increase the density, the ampere-turns being consumed by saturation of the iron, and even with a large decrease of m.m.f. the density is not decreased much, since with a small decrease 'I/ \\ L FIG. 97. Distribution of flux and m.m.f. at no load. of density the ampere-turns consumed by the saturation of the iron become available for the air gap. Thus, while in Fig. 95 the densities at the center and the two pole corners of the field pole are 8000, 12,000, and 4000, with the saturated structure in Fig. 99 they are 8000, 9040, and 6550. At or near the theoretical neutral, however, the saturation has no effect. That is, saturation of the armature teeth affords a means of 184 ELEMENTS OF ELECTRICAL ENGINEERING reducing the distortion of the magnetic field, or the shifting of flux at the pole corners, and is thus advantageous for machines which shall operate over a wide range of load with fixed position of brushes, if the brushes are shifted near to the next following pole corner. Fo J FIG. 98. Distribution of flux and m.m.f. at load, with Brushes at neutral. It offers no direct advantage, however, for machines corn- mutating with the brushes midway between the field poles, as converters. An effect similar to saturation in the armature teeth is produced J L FIG. 99. Distribution of flux and m.m.f. at load, with brushes shifted to next pole corner. by saturation of the field pole face, or more particularly, satura- tion of the pole corners of the field. VI. Effect of Commutating Poles 48. With the commutator brushes of a generator set midway between the field poles, as in Fig. 94, the m.m.f. of armature reac- D. C. COMMUTATING MACHINES 185 tion produces a magnetic field at the brushes. The e.m.f. gener- ated by the rotation of the armature through this field opposes the reversal of the current in the short-circuited armature coil under the brush, and thus impairs commutation. If therefore the commutation constants of the machines are not abnormally good high field strength, low armature reaction, low self-in- ductance and frequency of commutation the machine does not commutate satisfactorily under load, with the brushes midway between the field poles, and the brushes have to be shifted to the edge of the next field poles, as shown in Fig. 95, until the fringe of the magnetic flux of the field poles reverses the armature reac- tion and so generates an e.m.f. in the armature coil, which re- verses the current and thus acts as commutating flux. The commutating e.m.f. and therefore the commutating flux should be proportional to the current which is to be reversed, that is, to the load. The magnetic flux of the field pole of a shunt or compound machine, however, decreases with increasing load at the pole corners toward which the brushes are shifted, by the demagnetizing action of the armature reaction, and the shift of brushes therefore has to be increased with the load, from nothing at no load. At overload, the pole corners towarp! which the brushes are shifted may become so far weakened that even under the pole not sufficient reversing e.m.f. is generated, and satisfactory commutation ceases, that is, the sparking limit is reached. In general, however, varying the brush shift with the load is not permissible, and with rapidly fluctuating load not feasible, and therefore the brushes are set permanently at a mean shift. In this case, however, instead of increasing proportionally with the load, the commutating field is maximum at no load, and gradually decreases with increase of load, and is correct only at one particular load. At constant shift of the brushes, the commutation of the constant potential machine, direct-current generator or motor, is best at a certain load, and usually becomes poorer at lighter or heavier loads, and ultimately becomes bad by inductive sparks due to insufficient commutating flux. In machines in which very good commutating constants cannot be secured, as in large high-speed machines (steam turbine driven direct-current generators) , this may lead to bad sparking or even flashing over at sudden overloads as well as when throwing off full load. 186 ELEMENTS OF ELECTRICAL ENGINEERING 49. This has led to the development of the commutating pole, also called interpole, that is, a narrow magnetic pole located between the main poles at the point of the armature surface, at which commutation occurs, and excited so as to produce a commutating flux proportional to the load, and thus giving the required commutating field at all loads. Such machines then give no inductive sparking, but regarding commutation are limited in overload capacity only by the current density under the brush. Such commutating poles are excited by series coils, that is, coils connected in series with the armature and having a number of effective turns higher than the number of effective series turns per armature pole, so that at the position of the brushes the FIG. 100. Magnetic force distribution with commutating pole. m.m.f. of the commutating pole overpowers and reverses the m.m.f. of the armature, and produces a commutating m.m.f. equal to the product of the armature current and difference of commutating turns and armature turns, and thereby produces a commutating flux proportional to the load, as long as the mag- netic flux in the commutating poles does not reach too high magnetic saturation. In Fig. 100 is shown the distribution of m.m.f. around the cir- cumference of the armature, and in Fig. 101 the distribution of magnetic flux calculated in the manner as described in para- graphs 46 and 47. M represents the main poles, C the com- mutating poles. A main field excitation FQ is assumed of 10,000 ampere-turns per pole, and an armature reaction F a of 6000 D. C. COMMUTATING MACHINES 187 ampere-turns per pole. Choosing then 8000 ampere-turns per commutating pole F', leaves 2000 ampere-turns as resultant com- mutating m.m.f . at full load, half as much at half load, etc. The resultant m.m.f. of the main field F Q , the armature F a , and the commutating pole F f is represented in Fig. 100 by Fz, and the flux produced by it is shown in Fig. 101. As seen, with the com- mutator brushes midway between the field poles, that is, in the center of the commutating pole, a commutating flux proportional to the armature current enters the armature at the brush B and 5', and is cut by the revolving armature during commutation. The use of the commutating pole or interpole thus permits controlling the commutation, with fixed brush position midway between the field poles, and commutating poles therefore are FIG. 101. Magnetic flux distribution with commutating pole. extensively used in larger machines, especially of the high-speed type. The commutating pole makes the commutation independent of the main field strength, and therefore permits the machines to operate with equally good commutation over a wide voltage range, and at low voltage, that is, low field strength, as required for instance in boosters, etc. 50. With multiple-wound armatures, at least one commutat- ing pole for every pair of main poles is required, while with a series-wound armature a single commutating pole would be sufficient for all the sets of armature brushes, if of sufficient strength. In general, however, as many commutating poles as main poles are used. With the position of the brushes at the neutral, as is the case when using commutating poles, the armature reaction has no 188 ELEMENTS OF ELECTRICAL ENGINEERING demagnetizing component, and the only drop of voltage at load is that due to the armature resistance drop and the distortion of the main field, which at saturation produces a decrease of the total flux, as shown in Fig. 98. As is seen in Fig. 101, the magnetic flux of the commutating pole is not symmetrical, but the spread of flux is greater at the side of the main pole of the same polarity. As result thereof, the total magnetic flux is slightly increased by the commutating poles; that is, the two halves of the commutating flux on the two sides of the brush do not quite neutralize, and the com- mutating flux thus exerts a slight compounding action, that is, tends to raise the voltage. This can be still further increased by shifting the brushes slightly back and thus giving a magnet- izing component of armature reaction. This can be done with- out affecting commutation as long as the brushes still remain under the commutating pole. In this manner a compounding or even a slight over-compounding can be produced without a series winding on the main field poles, or, inversely, by shifting the brushes slightly forward, a demagnetizing component of armature reaction can be introduced. Furthermore, the current induced in the short-circuited armature coil by the commutating field is magnetizing, that induced by the magnetic field of arma- ture reaction, demagnetizing. In operating machines with commutating poles in multiple, care therefore must be taken not to have the compounding action of the commutating poles interfere with the distribution of load ; for this purpose an equalizer connection may be used between the commutating pole windings of the different machines, and the commutating windings treated in the same way as series coils on the main poles, that is, equalized between the different machines to insure division of load. 51. The advantage of the commutating pole over the shift of brushes to the edge of the next field pole, in constant poten- tial machines shunt or compound wound thus is that the commutating flux of the former has the right intensity at all loads, while that of the latter is right only at one particular load, too high below, too low above that load. In series-wound machines, that is, machines in which the main field is excited in series with the armature, and thus varies in strength with the armature current, armature reaction and field excitation are Always proportional to each other, and the distribution of mag- D. C. COMMUTATING MACHINES 189 netic flux at the armature circumference therefore always has the same shape, and its intensity is proportional to the current, except as far as saturation limits it. As the result thereof, shifting the brushes to the edge of the field poles, as in Fig. 95, brings them in a field which is proportional to the armature cur- rent and thus has the proper intensity as a commutating field. Therefore with series-wound machines commutating poles are not necessary for good commutation, but the shifting of the brushes gives the same result. However, in cases where the direc- tion of rotation frequently reverses, as in railway motors, the direction of the shift of brushes has to be reversed with the re- versal of rotation. In railway motors this cannot be done with- out objectionable complication, therefore the brushes have to be set midway, and the use of the magnetic flux at the edge of the next pole, as commutating flux, is not feasible. In this case a commutating pole is used, to give, without mechanical shifting of the brushes, the same effect which a brush shift would give. Therefore in railway motors, especially when wound for high voltage, as 1200 to 2400 volts, a commutating pole is sometimes used. This commutating pole, having a series winding just like the main pole, changes proportionally with the main pole. When reversing the direction of rotation, however, the armature and the commutating poles are reversed, while the main poles remain unchanged, or the main poles are reversed, while the arma- ture and the commutating poles remain unchanged; that is, the separate commutating pole becomes necessary because during the reversal of rotation it has to be treated differently from the main pole. 52. The commutating pole counteracts the armature reaction only at the place of commutation, but not elsewhere, and the field distribution resulting from the armature reaction thus is not eliminated by the commutating pole, except locally. Thus in machines having very low field excitation, and relatively high armature reaction, as alternating-current commutating machines, adjustable speed motors of wide speed range at the high-speed position, boosters near zero voltage, etc., the load losses resulting from excessive field distortion, the tendency to instability of speed, and the liability of flashing at the commutator at sudden changes of load are not eliminated by the commutating pole, but a more complete neutralization of the armature reaction is necessary. 190 ELEMENTS OF ELECTRICAL ENGINEERING Such is given by a compensating winding. This is a dis- tributed winding, located in the field pole faces closely adjacent to the armature, as shown in Fig. 102. It is connected in series but opposition to the armature winding, and of the same number of effective turns as the armature. By such a compensating winding, the armature reaction is completely eliminated, and with it magnetic distortion, load losses, etc. By giving the compensating winding some more ampere-turns than the armature, over-compensation is produced, giving a mag- netic cross flux under load, opposite to that of armature reaction, that is, a commutating flux. Very commonly in such com- pensated machines merely the ampere-turns of the compensat- ing winding in the slots at the commutating zone are increased, so that the compensating wind- ing all around the armature ex- actly neutralizes the armature reaction, except at the commu- tating zone, where it over-com- pensates and thus gives a local commutating flux. Such ma- chines, when properly designed, are characterized by absence of load losses, stability at all speeds, instant recovery at sudden load changes, and absence of sparking at commutator even at mo- mentary overloads of several hundred per cent. FIG. 102. Compensated com- mutating machine with fractional pitch armature winding. VII. Effect of Slots on Magnetic Flux 53. With slotted armatures the pole face density opposite the armature slots is less than that opposite the armature teeth, due to the greater distance of the air path in the former case. Thus, with the passage of the armature slots across the field pole a local pulsation of the magnetic flux in the pole face is produced, which, while harmless with laminated field pole faces, generates eddy currents in solid pole pieces. The frequency of this pul- sation is extremely high, and thus the energy loss due to eddy currents in the 'pole faces may be considerable, even with pul- sations of small amplitude. If S = peripheral speed of the arma- D. C. COMMUTATING MACHINES 191 ture in centimeters per second, l p = pitch of armature slot (that is, width of one slot and one tooth at armature surface), the S frequency is /i = y-. Or, if / = frequency of machine, n number of armature slots per pair of poles, /i = nf. For instance,/ = 33.3, n = 51, thus/i = 1700. Under the assumption, width of slots equals width of teeth = 2 X width of air gap, the dis- tribution of magnetic flux at the pole face is plotted in Fig. 103. The drop of density opposite each slot consists of two curved branches equal to those in Fig. 92, that is, calculated by B' -3 n FIG. 103. I < i slots on flu Iffect of B distribution. V + 1* 2 The average flux is 7525; that is, by cutting half the armature surface away by slots of a width equal to twice the length of air gap, the total flux under the field pole is reduced only in the proportion 8000 to 7525, or about 6 per cent. The flux B pulsating between 8000 and 5700 is equivalent to a uniform flux B\ = 7525 superposed with an alternating flux FIG. 104. Effect of slots on flux distribution. BO, shown in Fig. 104, with a maximum of 475 and a minimum of 1825. This alternating flux B Q can, as regards production of eddy currents, be replaced by the equivalent sine wave B o, that is, a sine wave having the same effective value (or square root of mean square). The effective value is 718. The pulsation of magnetic flux farther in the interior of the field-pole face can be approximated by drawing curves equi- 192 ELEMENTS OF ELECTRICAL ENGINEERING distant from B Q . Thus the curves #0.5, BI> ^1.5, #2, #2.5, and B 3 are drawn equidistant from B in the relative distances 0.5, 1, 1.5, 2, 2.5, and 3 (where l a = 1 is the length of air gap). They give the effective values: BQ BQ.S BI BI.Z BZ Bz.s B 3 718 373 184 119 91 69 57 That is, the pulsation of magnetic flux rapidly disappears toward the interior of the magnet pole, and still more rapidly the energy loss by eddy currents, which is proportional to the square of the magnetic density. 54. In calculating the effect of eddy currents, the magnetizing effect of eddy currents may be neglected (which tends to reduce the pulsation of magnetism); this gives the upper limit of loss Let B = effective density of the alternating magnetic flux, S = peripheral speed of armature in centimeters per second, and I = length of pole face along armature. The e.m.f. generated in the pole face is then e = SIB X 10- 8 , and the current in a strip of thickness Al and 1 cm. width, eAl SIBAl 10~ 8 SBAl 10~ 8 Ai = r = - jr - = - i pl> pi p where p = resistivity of the material. Thus the effect of eddy currents in this strip is ' SHB 2 Al 10- 16 Ap = eAi = - i or per cubic centimeter, S*B* 10~ 16 P = ~~P~ that is, proportional to the square of the effective value of mag- netic pulsation, the square of peripheral speed, and inversely proportional to the resistivity. Thus, assuming for instance, S = 2000, p = 20 X 10~ 6 , for cast steel, p = 100 X 10~ 6 , for cast iron, we have in the above example, D. C. COMMUTATING MACHINES 193 At distance from pole face B p Cast steel Cast iron 718 10.3 2.06 la 2 373 2.78 0.56 la 184 0.677 0.135 ~2~ 119 0.283 0.057 2 la 91 0.166 0.033 5 la 2 69 0.095 0.019 3 la 57 0.065 0.013 VIII. Armature Reaction 55. At no load, that is, with no current in the armature cir- cuit, the magnetic field of the commutating machine is sym- metrical with regard to the field poles. Thus the density at the armature surface is zero at the point or in the range midway between adjacent field poles. This point, or range, is called the "neutral" point or "neutral" range of the commutating machine. Under load the armature current represents a m.m.f. acting in the direction from commutator brush to commutator brush of opposite polarity, that is, in quadrature with the field m.m.f. if the brushes stand midway between the field poles; or shifted against the quadrature position by the same angle by which the commutator brushes are shifted, which angle is called the angle of lead. If n = turns in series between brushes per pole, and i = cur- rent per turn, the m.m.f. of the armature is F a = ni per pole. Or, if r?o = total number of turns on the armature, n c = number of turns or circuits in multiple, 2n p = number of poles, and t' = total armature current, the m.m.f. of the armature per pole is F a = ^ This m.m.f. is called the armature reaction of the 2n p n c continuous-current machine. Since the armature turns are distributed over the total pitch of pole, that is, a space of the armature surface representing 180 deg., the resultant armature reaction is found by multiplying 194 ELEMENTS OF ELECTRICAL ENGINEERING C -j- go 2 F a with the average cos = , and is thus yu ^" F ao 2 F a 2 ni When comparing the armature reaction of commutating ma- chines with other types of machines, as synchronous machines 2 F a etc., the resultant armature reaction F ao = - - has to be used. In discussing commutating machines proper, however, the value F a = ni is usually considered as the armature reaction. 56. The armature reaction of the commutating machine has a distorting and a magnetizing or demagnetizing action upon the magnetic field. The armature ampere-turns beneath the field poles have a distorting action as discussed under " Magnetic Dis- tribution" in the preceding paragraphs. The armature ampere- turns between the field poles have no effect upon the resultant field if the brushes stand at the neutral; but if the brushes are shifted, the armature ampere-turns inclosed by twice the angle of lead of the brushes have a demagnetizing action. Thus, if r = pole arc as fraction of pole pitch, TI = shift of brushes as fraction of pole pitch, F a the m.m.f. of armature reaction, and FQ the m.m.f. of field excitation per pole, the demag- netizing component of armature reaction is riF a , the distorting component of armature reaction is rF a , and the magnetic density at the strengthened pole corner thus corresponds to the m.m.f. rF a rFa FQ + -- at the weakened field corner to the m.m.f. FQ g IX. Saturation Curves 57. As saturation curve or magnetic characteristic of the com- mutating machine is understood the curve giving the generated voltage, or terminal voltage at open circuit and normal speed, as function of the ampere-turns per pole field excitation. Such curves are of the shape shown in Fig. 105 as A. Owing to the remanent magnetism or hysteresis of the iron part of the magnetic circuit, the saturation curve taken with decreasing field excitation usually does not coincide with that taken with increasing field excitation, but is higher, and by gradually first increasing the field excitation from zero to maximum and then decreasing again, the looped curve in Fig. 106 is derived, giving D. C. COMMUTATING MACHINES 195 as average saturation curve the curve shown in Fig. 105 as A and as central curve in Fig. 106. Direct-current generators are usually operated at a point of the saturation curve above the bend, that is, at a point where the terminal voltage increases considerably less than proportionally to the field excitation. This is necessary in self-exciting direct- current generators to secure stability. The ratio increase of field excitation total field excitation that is, corresponding increase of voltage total voltage F* de FIG. 105. Saturation characteristics. is called saturation factor s, and is plotted in Fig. 105. It is the ratio of a small percentage increase in field excitation to a corre- sponding percentage increase in voltage thereby produced. The quantity 1 is called the percentage saturation of the ma- s chine, as it shows the approach of the machine field to mag- netic saturation. 58. Of considerable importance also are curves giving the terminal voltage as function of the field excitation at load. Such curves are called load saturation curves, and can be constant 196 ELEMENTS OF ELECTRICAL ENGINEERING current load saturation curve, that is, terminal voltage as func- tion of field ampere-turns at constant full-load current through the armature, and constant resistance load saturation curve, that is, terminal voltage as function of field ampere-turns if the machine circuit is closed through a constant resistance giving full-load current at full-load terminal voltage. A constant current load saturation curve is shown as B, and a constant resistance load saturation curve as C in Fig. 105. FIG. 106. Saturation curves. X. Compounding 59. In the direct-current generator the field excitation re- quired to maintain constant terminal voltage has to be increased with the load. A curve giving the field excitation in ampere- turns per pole, as function of the load in amperes, at constant terminal voltage, is called the compounding curve of the machine. The increase of field excitation required with load is due to : 1. The internal resistance of the machine, which consumes e.m.f. proportional to the current, so that the generated e.m.f., and thus the field m.m.f. corresponding thereto, has to be greater under load. If p = resistance drop in the machine as fraction ir of terminal voltage, = > the generated e.m.f. at load has to be e (1 + p), and if ^o= no-load field excitation, and s = satu- D. C. COMMUTATING MACHINES 197 ration coefficient, the field excitation required to produce the e.m.f . e (1 + p) is Fo (1 + sp) ; thus an additional excitation of spF is required at load, due to the armature resistance. 2. The demagnetizing effect of the ampere-turns armature reaction of the angle of shift of brushes TI requires an increase of field excitation by riF a . (Section VII.) 3. The distorting effect of armature reaction does not change the total m.m.f. producing the magnetic flux. If, however, mag- netic saturation is reached or approached in a part of the mag- netic circuit adjoining the air gap, the increase of magnetic density at the strengthened pole corner is less than the decrease at the weakened pole corner, and thus the total magnetic flux with the same total m.m.f. is reduced, and to produce the same total magnetic flux an increased total m.m.f., that is, increase of field excitation, is required. This increase depends upon the saturation of the magnetic circuit adjacent to the armature conductors. 4. The magnetic stray field of the machine, that is, that part of the magnetic flux which passes from field pole to field pole with- out entering the armature, usually increases with the load. This stray field is proportional to the difference of magnetic potential between field poles; that is, at no load it is proportional to the ampere-turns m.m.f. consumed in air gap, armature teeth, and armature core. Under load, with the same generated e.m.f., that is, the same magnetic flux passing through the armature core, the difference of magnetic potential between adjacent field poles is increased by the counter m.m.f. of the armature and by saturation. Since this magnetic stray flux passes through field poles and yoke, the magnetic density therein is increased and the field excitation correspondingly, especially if the magnetic den- sity in field poles and yoke is near saturation. This increase of field strength required by the increase of density in the external magnetic circuit, due to the increase of magnetic stray field, depends upon the shape of the magnetic circuit, the armature reaction, and the saturation of the external magnetic circuit. Curves giving, with the amperes output as abscissas, the ampere-turns per pole field excitation required to increase the voltage proportionally to the current are called over-compounding curves. In the increase of field excitation required for over- compounding, the effects of magnetic saturation are still more marked. 13 198 ELEMENTS OF ELECTRICAL ENGINEERING XL Characteristic Curves 60. The field characteristic or regulation curve, that is, curve giving the terminal voltage as function of the current output at constant field excitation, is of less importance in commutating machines than in synchronous machines, since commutating machines are usually not operated with separate and constant excitation, and the use of the series field affords a convenient means of changing the field excitation proportionally to the load. The curve giving the terminal voltage as function of current out- put, in a compound-wound machine, at constant resistance in the shunt field, and constant adjustment of the series field, is, how- ever, of importance as regulation curve of the direct-current generator. This curve would be a straight line except for the effect of saturation, etc., as discussed above. XII. Efficiency and Losses 61. The losses in a commutating machine which have to be considered when deriving the efficiency by adding the individual losses are: 1. Loss in the resistance of the armature, the commutator leads, brush contacts and brushes, in the shunt field and the series field with their rheostats. 2. Hysteresis and eddy currents in the iron at a voltage equal to the terminal voltage, plus resistance drop in a generator, or minus resistance drop in a motor. 3. Eddy currents in the armature conductors when large and not protected, and in pole faces when solid and the air gap is small. 4. Friction of bearings, of brushes on the commutator, and windage. 5. Load losses, due to the increase of hysteresis and of eddy currents under load, caused by the change of the magnetic dis- tribution, as local increase of magnetic density and of stray field. The friction of the brushes and the loss in the contact resist- ance of the brushes are frequently quite considerable, especially with low-voltage machines. Constant or approximately constant losses are: friction of bearings and of commutator brushes, and windage; hysteresis and eddy current losses; and shunt field excitation. Losses D.'C. COMMUTATING MACHINES 199 increasing with the load, and proportional or approximately proportional to the square of the current: armature resistance losses; series field resistance losses; brush contact resistance losses; and the so-called "load losses." XIII. Commutation 62. The most important problem connected with commutating machines is that of commutation. Fig. 107 represents diagrammatically a commutating machine. FIG. 107. Diagram for the study of commutation. The e.m.f. generated in an armature coil A is zero with this coil at or near the position of the commutator brush B\. It rises and reaches a maximum about midway between two adjacent sets of brushes, BI and B 2 , at C, and then decreases again, reaching zero at or about B 2 , and then repeats the same change in opposite direction. The current in armature coil A, however, is constant during the motion of the coil from BI to BI. While the coil A passes the brush B 2 , however, the current in the coil A reverses, and then remains constant again in opposite direc- 200 ELEMENTS OF ELECTRICAL ENGINEERING tion during the motion from B 2 to B 3 . Thus, while the armature coils of a commutating machine are the seat of a system of poly- phase e.m.fs. having as many phases as coils, the current in these coils is constant, reversing successively. 63. The reversal of current in coil A takes place while the gap G between the two adjacent commutator segments between which the coil A is connected passes the brush B 2 . Thus, if l w = width of brushes, S = peripheral speed of commutator per second in the same measure in which l w is given, as in inches per second if Z is given in inches, to = - is the time during which the current in A reverses. Thus, considering the reversal as a 1 S single alternation, t Q is a half period, and thus / = ^-7- = ;ry- is 4 o z i w the frequency of commutation; hence, if L = inductance of the armature coil A, the e.m.f. generated in the armature coil during commutation is eo = 2irfoLio t where io = current reversed, and the energy which has to be dissipated during commutation is i and, solving the exponential equation for e, we obtain D. C. COMMUTATING MACHINES 203 It is evident that the inequality e > i<>r must be true, otherwise perfect commutation is not possible. If we have that is, the current never reverses, but merely dies out more or less, and in the moment when the gap G of the armature coil leaves the brush B the current therein has to rise suddenly to full intensity in opposite direction. This being impossible, due to the inductance of the coil, the current forms an arc from the brush across the commutator surface for a length of time depend- ing upon the inductance of the armature coil. Therefore, with low-resistance brushes, resistance commutation is not permissible except with machines of extremely low arma- FIG. 108. Brush commutating coil A. ture inductance, that is, armature inductance so low that the magnetic energy -7^, which appears as "spark" in this case, is & harmless. Voltage commutation is feasible with low-resistance brushes, but requires a commutating e.m.f. e proportional to current z'o; that is, requires shifting of brushes proportionally to the load, or a commutating pole. In the preceding, the e.m.f. e.has been assumed constant dur- ing the commutation. In reality it varies somewhat, usually increasing with the approach of the commutated coil to a denser field. It is not possible to consider this variation in general, and e is thus to be considered the average value during commutation. 66. (b) High-resistance brush contact. Fig. 108 represents a brush B commutating armature coil A. 204 ELEMENTS OF ELECTRICAL ENGINEERING Let r = contact resistance of the brush, that is, resistance from the brush to the commutator surface over the total bearing surface of the brushes. The resistance of the commutated cir- cuit is thus internal resistance of the armature coil r plus the resistance from C to B plus the resistance from B to D. Thus, if to = time of commutation, at the time t after the be- ginning of the commutation, the resistance from C to B is and from B to D is -; thus, the total resistance of the corn- to * mutated coil is T- , to?*0 i 'o7*0 to TQ R = r + + -. : = r + ( v t to t t (to t) If i = current in coil A before commutation, the total cur- rent into the armature from brush B is 2 i . Thus, if i current in commutated coil, the current from B to D = i Q + i, the cur- rent from B to C = io i. Hence, the difference of potential from D to C is The e.m.f. acting in coil A is Ldi and herefrom the difference of potential from D to C is L-- > hence, di . tofo , . .>. tofo / . .\ dt to t t or, transposing, Ldi . toToio (2 1 to) to z roi dt t (to t) t (to t) T di . / roto 2 \ Totoio C2>t to) ' t(t Q -t) * The further solution of this general problem becomes difficult, but even without integrating this differential equation a number of important conclusions can be derived. Obviously the commutation is correct and thus sparkless, if D. C. COMMUTATING MACHINES 205 the current entering over the brush shifts from segment to seg- ment in direct proportion to the motion of the gap between ad- jacent segments across the brush, that is, if the current density is uniform all over the contact surface of the brush. This means that the current i in the short-circuited coil varies from + io to i Q as a linear function of the time. In this case it can be rep- resented by . . to-2t ^ = ^o r J to thus, di = 2ip dt~ to * Substituting this value in the general differential equation gives, after some transformation, -?-*<> + r(to - 20 - 2L = 0; or, e = i I which gives at the beginning of commutation, t = 0, at the end of commutation, t = t Q , that is, even with high-resistance brushes, for perfect com- mutation, voltage commutation is necessary, and the e.m.f. e impressed upon the commutated coil must increase during com- mutation from ei to 6 2 , by the above equation. This e.m.f. is proportional to the current i Q> but is independent of the brush resistance r . RESISTANCE COMMUTATION 67. Herefrom it follows that resistance commutation cannot be perfect, but that at the contact with the segment that leaves the brush the current density must be higher than the average. Let g = ratio of actual current density at the moment of leaving the brush to average current density of brush contact, and con- 20G ELEMENTS OF ELECTRICAL ENGINEERING sidering only the end of commutation, as the most important moment, we have . . (2g -l)U-2gt I lQ -- - -- to For t = to t l this gives ^1 i = - io + 2 g - io, to while uniform current density would require ^1 i = io + 2 - i . to The general differential equation of resistance commutation, e = 0, is di rplo 2 \ r Q t i (2t- t ) Substituting in this equation the value of i from the foregoing equation, expanding and cancelling to t, we obtain 2 r *o 2 (g - 1) +. rtt (2 g - 1) - 2 grt 2 - 2 gLt = 0; hence, f rt) g ~ 2 (r * 2 + rtto - rt* - Lt) and for t = t , 2(r Q t Q -L) ~r Q t Q -L' that is, g is always > 1. The smaller L and the larger r Q) the smaller is g; that is, the nearer it is to 1, the condition of perfect commutation, and the better is the commutation. Sparkless commutation is impossible for very large values of g, that is, when L approaches roto, or when r is not much larger than For this reason, in machines in which L cannot be o made small, r is sometimes made large by inserting resistors in the leads between the armature and the commutator, so-called ''resistance" or "preventive" leads as used in alternating-current commutator motors. XIV. Types of Commutating Machines 68. By the methods of excitation, commutating machines are subdivided into magneto, separately excited, shunt, series, D. C. COMMUTATING MACHINES 207 and compound machines. Magneto machines and separately excited machines are very similar in their characteristics. In either, the field excitation is of constant, or approximately constant, impressed m.m.f. Magneto machines, however, are little used, except for very small sizes. By the direction of energy transformation, commutating ma- chines are subdivided into generators and motors. Of foremost importance in discussing the different types of machines is the saturation curve or magnetic characteristic; that is, a curve relating terminal voltage at constant speed to ampere-turns per pole field excitation, at open circuit. Such a curve is shown as A in Figs. 109 and 110. It has the same 1 8 9 4 5 6 78 9 19 U 12 13 H 15 16 I? J W 20 81 FIG. 109. Generator saturation curves. general shape as the magnetic flux density curve, except that the knee or bend is less sharp, due to the different parts of the magnetic circuit saturation successively. Thus, in order to generate voltage ac the field excitation oc is required. Subtracting from ac in a generator, Fig. 109, or adding in a motor, Fig. 110, the value ab = ir, the voltage con- sumed by the resistance of the armature, commutator, etc., gives the terminal voltage be at current i, and adding to oc the value ce = bd = iq = armature reaction, or rather field excita- tion required to overcome the armature reaction, gives the field excitation oe required to produce the terminal voltage de at 208 ELEMENTS OF ELECTRICAL ENGINEERING current i. The armature reaction iq, corresponding to current i, is calculated as discussed before, and q may be called the coef- ficient of armature reaction. 69. Such a curve, D, shown in Fig. 109 for a generator, and in Fig. 110 for a motor, and giving the terminal voltage de at current i, corresponding to the field excitation oe } is called a load saturation curve. Its points are respectively distant from the corresponding points of the no-load saturation curve A a constant distance equal to ad, measured parallel thereto. Curves D are plotted under the assumption that the armature reaction is constant. Frequently, however, at lower voltage the 1231567 FIG. 110. 9 10 11 12 13 14 .15 16 17 18 19 20 21 -Motor saturation curves. armature reaction, or rather the increase of excitation required to overcome the armature reaction iq, increases, since with voltage commutation at lower voltage, and thus weaker field strength, the brushes have to be shifted more to secure spark- less commutation, and thus the demagnetizing effect of the angle of lead increases. At higher voltage iq usually increases also, due to increase of magnetic saturation under load, caused by the increased stray field. Thus, the load saturation curve of the continuous-current generator more or less deviates from the theoretical shape D toward a shape shown as G. D. C. COMMUTATING MACHINES A. GENERATORS 209 Separately Excited and Magneto Generator 70. In a separately excited or magneto machine, that is, a machine with constant field excitation F Q) a demagnetization \ \\ 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 FIG. 111. Separately excited or magneto-generator demagnetization curve and load characteristic with constant shift of brushes. 10 20 30 40 50 60 70 80 90 100 110 120 130 FIG. 112. Separately excited or magneto-generator demagnetization curve and load characteristic with variable shift of brushes. curve can be plotted from the magnetization or saturation curve A in Fig. 109. At current i, the resultant m.m.f . of the machine is F Q iq, and the generated voltage corresponds thereto by the saturation curve A in Fig. 110. Thus, in Fig. Ill a de- magnetization curve A is plotted with the current ob = i as 210 ELEMENTS OF ELECTRICAL ENGINEERING abscissas and the generated e.m.f. ab as ordinates, under the assumption of constant coefficient of armature reaction q, that is, corresponding to curve D in Fig. 109. This curve becomes zero at the current ?o, which makes i$q = FQ. Subtracting from curve A in Fig. Ill the drop of voltage in the armature and commutator resistance, ac = ir, gives the external characteristic B of the machine as generator, or the curve relating the terminal voltage to the current. In Fig. 112 the same curves are shown under the assumption that the armature reaction varies with the voltage in the way as represented by curve G in Fig. 109. In a separately excited or magneto motor at constant speed the external characteristic would lie as much above the demag- netization curve A as it lies below in a generator in Fig. Ill, and at constant voltage the speed would vary inversely pro- portional hereto. Shunt Generator 71. The external or load characteristic of the shunt generator is plotted in Fig. 113 with the current as abscissas and the terminal voltage as ordinates, as A for constant coefficient of armature reaction, and as B for a coefficient of armature reac- tion varying with the voltage in the way as shown in G, Fig. 109. The construction of these curves is as follows: In Fig. 109, og is the straight line giving the field excitation oh as function of the terminal voltage hg (the former obviously being proportional to the latter in the shunt machine). The open-circuit or no-load voltage of the machine is then kq. Drawing gl parallel to da (assuming constant coefficient of armature reaction, or parallel to the hypothenuse of the triangle iq, ir at voltage og, when assuming variable armature reaction), then the current which gives voltage gh is proportional to gl, that is, i : i Q = gl : da, where i Q is the current at the voltage de. As seen from Fig. 113, a maximum value of current exists which is less if the brushes are shifted than at constant position of brushes. From the load characteristic of the shunt generator the resistance characteristic is plotted in Fig. 114; that is, the de- pendence of the terminal voltage upon the external resistance terminal voltage ~ . ^. , R = *- Curve A in Fig. 114 corresponds to current D. C. COMMUTATING MACHINES 211 constant, curve B to varying armature reaction. It is seen that at a certain definite resistance the voltage becomes zero, and for lower resistance the machine cannot generate but loses its excitation. The variation of the terminal voltage of the shunt generator with the speed at constant field resistance is shown in Fig. 115, at no load as A, and at constant current i as B. These curves are derived from the preceding ones. They show that below a certain speed, which is much higher at load than at no load, the r 50 100 150 200 250 300 350 FIG. 113. Shunt generator load characteristic. machine cannot generate, and cannot be realized. The lower part of curve B is unstable Series Generator 72. In the series generator the field excitation is proportional to the current i, and the saturation curve A in Fig. 116 can thus be plotted with the current i as abscissas. Subtracting ab = ir, the resistance drop, from the voltage, and adding bd = iq, the armature reaction, gives a load saturation curve or external characteristic B of the series generator. The terminal voltage is zero at no load or open circuit, increases with the load, reaches a maximum value at a certain current, and then decreases again and reaches zero at a certain maximum current, the current of short circuit. Curve B is plotted with constant coefficient of armature reac- tion q. Assuming the brushes to be shifted with the load and 212 ELEMENTS OF ELECTRICAL ENGINEERING 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.1 1.6 1.8 2.0 FIG. 114. Shunt generator resistance characteristic. / 1RO / 170 / / A 160 150 140 130 / / / / / ' / / / / / no / t / 100 / ' 90 80 70 60 50 40 30 A / 4 I / / / v / *- A / / SO ( 10 / SPEE -> I 0.2 0.4 0.6 0.8 1.0 1.2 1.4 FIG. 115. Shunt generator speed characteristic at constant field circuit resistance. D. C. COMMUTATING MACHINES 213 proportionally to the load, gives curves C, D, and E, which are higher at light load, but fall off faster at high load. A still further shift of brushes near the maximum current value even overturns the curve as shown in F. Curves E and F correspond to a very great shift of brushes, and an armature demagnetizing effect of the same magnitude as the field excitation, as realized in arc-light machines, in which the last part of the curve is used to secure inherent regulation for constant current. The resistance characteristic, that is, the dependence of the current and of the terminal voltage of the series generator upon 6000 6000 1 23 i 5 6 7 8 9 10 11 12 13 U 15 16 17 18 19 FIG. 116. Series generator saturation curve and load characteristic. the external resistance, is constructed from Fig. 116 and plotted in Fig. 117. BI and B z in Fig. 117 are terminal volts and amperes corre- sponding to curve B in Fig. 116, #1, E z , and F% volts and amperes corresponding to curves E and F in Fig. 116. Above a certain external resistance the series generator loses its excitation, while the shunt generator loses its excitation below a certain external resistance. Compound Generator 73. The saturation curve or magnetic characteristic A, and the load saturation curves D and G of the compound generator, are shown in Fig. 118 with the ampere-turns of the shunt field 214 ELEMENTS OF ELECTRICAL ENGINEERING as abscissas. A is the same curve as in Fig. 109, while D and G in Fig. 118 are the corresponding curves of Fig. 109 shifted to 7000 50001 8000 1000 \ RE ISTA.NCE ANCE \ 200 400 600 800 1000 1200 1400 1600 180 FIG. 117. Series generator resistance characteristic. ;>OQO OMMS 400 3000 1000 5000 6000 7000 8000 9000 FIG. 118. Compound generator saturation curve. the left by the distance iq Q , the m.m.f. of ampere-turns of the series field. At constant position of brushes the compound generator, when D. C. COMMUTATING MACHINES 215 adjusted for the same voltage at no load and at full load, under- compounds at higher and over-compounds at lower voltage, and even at open circuit of the shunt field gives still a voltage op as series generator. When shifting the brushes under load, at lower voltage a second point g is reached where the machine compounds correctly, and below this point the machine under-compounds and loses its excitation when the shunt field decreases below a certain value; that is, it does not excite itself as series generator. B. MOTORS Shunt Motor 74. Three speed characteristics of the shunt motor at con- stant impressed e.m.f. e are shown in Fig. 116 as A, P, Q, corre- sponding to the points d, p, q of the motor load saturation curve, Fig. 110. Their derivation is as follows: At constant impressed ir,0 1100 WOO 00 (00 w 50 100 150 200250 300 350 400 450 500 FIG. 119. Shunt motor speed curves, constant impressed e.m.f. e.m.f. e the field excitation is constant and equals F Q , and at current i the generated e.m.f. must be e ir. The resultant field excitation is F iq t and corresponding hereto at constant speed the generated e.m.f. taken from saturation curve A in Fig. 110 is e\. Since it must be e ir, the speed is changed in , . e - ir the proportion At a certain voltage the speed is very nearly constant, the demagnetizing effect of armature reaction counteracting the effect of armature resistance. At higher voltage the speed falls, at lower voltage it rises with increasing current. In Fig. 120 is shown the speed characteristic of the shunt 216 ELEMENTS OF ELECTRICAL ENGINEERING motor as function of the impressed voltage at constant output, that is, constant product, current times generated e.m.f. If i = current and P = constant output, the generated e.m.f. p must be approximately e\ = , and thus the terminal voltage e = e\ + ir. Proportional hereto is the field excitation FQ. The resultant m.m.f. of the field is thus F = F Q iq, and corre- sponding thereto from curve A in Fig. Ill is derived the e.m.f. e Q which would be generated at constant speed by the m.m.f. F. Since, however, the generated e.m.f. must be e\ } the speed is changed in the proportion 00 REV. PER 2000 1800 1600 1400 1200 1000 800 COO [400 200 / * / / / s / / / 7 nt / / 1 / / 1 / ^r S vor -s* 20 10 60 80 100 120 110 160 ISO 200 FIG. 120. Shunt motor speed curve, variable impressed e.m.f. The speed rises with increasing and falls with decreasing im- pressed e.m.f. Still further decreasing the impressed e.m.f., the speed reaches a minimum and then increases again, but the conditions become unstable. Series Motor 75. The speed characteristic of the series motor is shown in Fig. 121 at constant impressed e.m.f. e. A is the saturation curve of the series machine, with the current as abscissas and at constant speed. At current i, the generated e.m.f. must be /? A f e ir, and the speed is thus '- times that, for which curve A Q\ is plotted, where e\ = e.m.f. taken from saturation curve A. D. C. COMMUTATING MACHINES 217 This speed curve corresponds to a constant position of brushes midway between the field poles, as generally used in railway motors and other series motors. If the brushes have a constant shift or are shifted proportionally to the load, instead of the saturation curve A in Fig. 121 a curve is to be used correspond- ing to the position of brushes, that is, derived by adding to the abscissas of A the values iq, the demagnetizing effect of arma- ture reaction. 10 60 FIG. 121 100 120 _110 160 ISO -Series motor speed curve. The torque of the series motor is shown also in Fig. 121, derived as proportional to A X i, that is, current X magnetic flux. Compound Motors 76. Compound motors can be built with cumulative com- pounding and with differential compounding. Cumulative compounding is used to a considerable extent, as in elevator motors, etc., to secure economy of current in starting and at high loads at the sacrifice of speed regulation; that is, a compound motor with cumulative series field stands in its speed and torque characteristic intermediate between the shunt motor and the series motor. 218 ELEMENTS OF ELECTRICAL ENGINEERING Differential compounding is used to secure constancy of speed with varying load, but to a small extent only, since the speed regulation of a shunt motor can be made sufficiently close, as was shown in the preceding. Conclusion 77. The preceding discussion of commutating machine types can obviously be only very general, showing the main character- istics of the curves, while the individual curves can be modified to a considerable extent by suitable design of the different parts of the machine when required to derive certain results, as, for instance, to extend the constant-current part of the series gen- erator; or to derive a wide range of voltage at stability, that is, beyond the bend of the saturation curve in the shunt generator; or to utilize the range of the shunt generator load characteristic at the maximum current point for constant-current regulation; or to secure constancy of speed in a shunt motor at varying impressed e.m.f., etc. The use of the commutating machine as direct-current con- verter has been omitted from the preceding discussion. By means of^ one or more alternating-current compensators or autotransformers, connected to the armature by collector rings, the commutating machine can be used to double or halve the voltage, or convert from one side of a three-wire system to the other side and, in general, to supply a three-wire Edison system from a single generator. Since, however, the direct- current converter and three-wire generator exhibit many fea- tures similar to those of the synchronous converter, as regards the absence of armature reaction, the reduced armature heat- ing, etc., they will be discussed as an appendix to the synchro- nous converter. XV. APPENDIX ALTERNATING-CURRENT COMMUTATOR MOTOR 78. Since in the series motor and in the shunt motor the direction of the rotation remains the same at a reversal of the impressed voltage, these motors can be operated by an alternat- ing voltage, as alternating-current motors, by making such changes in the materials, proportioning and design, as the al- ternating nature of the current requires. D. C. COMMUTATING MACHINES 219 In the alternating-current commutator motor, the field struc- ture as well as the armature must be laminated, since the mag- netic flux is alternating. The alternation of the field flux induces an e.m.f. of self induction in the field winding. In the shunt motor, this causes the field exciting current and with it the magnetic field flux to lag and thereby to be out of phase with the armature current which, to represent work, must essentially be an energy current, and thereby reduces output and efficiency and hence requires some method of compensation, as capacity in series with the field winding or excitation of the field from a quadrature phase of voltage. In the series motor the self-inductance of the field causes the main current to lag behind the impressed voltage and thereby lowers the power-factor of the motor. Thus, to get good power-factor, the field self-inductance must be made low, that is, the field as weak and the armature as strong as possible. With such a strong armature, and weak field, the commutating pole is not sufficient to control magnetic distortion by the arma- ture reaction, and complete compensation by a distributed compensating winding, as Fig. 102, page 190, is required. 79. When in the position of commutation the armature coil is short-circuited by the commutator brush, it encloses the full field flux and thus for a moment no e.m.f. is induced in the armature coil by its rotation through the field flux, and in the continuous current machine the coil is without voltage except whatever voltage may be intentionally produced by the com- mutating flux. In the alternating-current motor, however, the field flux induces voltage also in the armature coil by its alternation, and this voltage is a maximum in the position of commutation, and when short-circuited by the commutator brush tends to produce an excessive current and cause spark- ing. No position exists on the commutator of the alternating- current motor where the armature coil does not contain an induced e.m.f., but in the position midway between the brushes the e.m.f. induced by the rotation through the magnetic field is a maximum; in the position of commutation the e.m.f. induced by the alternation of the field flux is a maximum. To overcome the destructive sparking caused by the short circuit of the latter e.m.f. by the commutator brush is the problem of making a successful alternating-current commutator: 1. Inducing an opposite e.m.f. by a commutating field. As 220 ELEMENTS OF ELECTRICAL ENGINEERING the e.m.f. induced by the alternation of the main field is in quadrature with the main field, and the e.m.f. induced by the rotation through the commutating field is in phase with it, the commutating field must be in quadrature with the main field. By properly proportioning this commutating field, as in the series repulsion motor, completely sparkless commutation can be produced at speed. However, at standstill and low speeds this method fails, as the voltage induced by the rotation through the commutating field becomes zero at standstill. 2. Reducing the short-circuit current by high resistance leads between commutator and armature coil. This only mitigates the trouble, but due to the voltage drop in the lead resistance tends to increase sparking at speed. Also, the excessive con- centration of heat in the commutating leads in the moment of starting tends to destroy them if the motor does not quickly start. 3. Narrow brushes, to reduce the duration of short circuit. 4. Low impressed frequency, so as to give low values to the induced e.m.f. This is the cause of the desire for abnormally low frequencies, as 15 and even 8 cycles, in alternating-current railway electrification. 5. Low magnetic flux per pole. This is the reason why alternating-current commutator motors of large power usually have such a large number of poles. These very severe limitations of the design of alternating-cur- rent commutating motors are the reason why such motors have found only limited application, except in smaller sizes. 80. Alternating-current motors are usually single-phase, since the possibility of commutation control makes the single-phase easier than a polyphase design. In the single-phase motor, the magnetic field flux is constant in direction, and the direction in quadrature to the main field flux thus is available for pro- ducing a suitable commutating flux. In the polyphase motor, however, the magnetic flux rotates, assuming successively all directions, and thus no commutating flux can be used. For this reason, designs of polyphase commutator motors have been made in which the different (2 and 3) phases are kept separate, and spaces left between them for accommodating commutating fluxes. 81. Alternating-current commutator motors are used: 1. In railroading, for securing the advantage of the higher D. C. COMMUTATING MACHINES 221 economy of high voltage alternating-current transmission and distribution. For railroading generally the series motor type is used, either the plain compensated series motor, or inductive modifications thereof, as the repulsion motor etc. In the repul- sion motor the armature, instead of being connected in series with field and compensating winding, is closed on itself and thus traversed by a secondary current induced by the compensating winding as primary that is, the armature is connected inductively in series. 2. As constant-speed motor where considerable starting torque is required, as for elevators, hoists, etc., and in general as self-starting single-phase motors. For this purpose, com- binations of repulsion and induction type or of series and in- duction type are used. 3. As adjustable speed, alternating-current motor of single- phase and of polyphase type. The synchronous motor and the induction motor both are constant and fixed speed, the former synchronous, the latter near synchronous. Operating the induction motor materially below synchronism, by arma- ture resistance, is inefficient and gives a speed which varies with the load. By changing the number of poles, or by concatena- tion, multi-speed induction motors can be produced. The gradual speed adjustment, as given by field control of direct- current motors, requires, however, a commutator on the al- ternating-current motor. If into the secondary of the induction motor an e.m.f. is introduced, the speed of the motor can be varied by varying the introduced e.m.f.; and lowered, if this e.m.f. is in opposition; raised beyond synchronism, if this e.m.f. is in the same direction as the e.m.f. induced in the motor secondary. As, however, the e.m.f. induced in the induction motor secondary is of the frequency of slip, the speed controlling e.m.f. must either be supplied through the commutator or de- rived from a low frequency commutating machine as source. 4. For power-factor compensation. In an inductive circuit, the current lags behind the voltage or, what is the same, the voltage leads the current, and the power-factor thus can be raised by compensation either by introducing a leading current, as from condenser or overexcited synchronous motor, or by in- troducing a lagging voltage. In the commutating machines, the voltage induced in the armature by its rotation is in phase with the field magnetism, and by lagging the field exciting current, 222 ELEMENTS OF ELECTRICAL ENGINEERING the commutating machines thus can be made to give a lagging voltage, that is, to compensate for low power-factor due to lagging current. Thus, by inserting such a commutating machine into the secondary of an induction machine, the latter can be made to give unity power-factor or even leading current. Such phase compensation is frequently used in alternating- current commutator motors to get good power-factor. Thus in the series motor, by shunting the field by a non-inductive re- sistance, and thereby lagging the field exciting component of the current and with it the field flux and the voltage induced in the armature by its rotation, behind the main current, the series motor can at higher speeds be made to give unity power- factor. At low speeds, such complete compensation is not possible, as the compensating voltage is proportional to the speed. C. SYNCHRONOUS CONVERTERS I. General 82. For long-distance transmission, and to a certain extent also for distribution, alternating currents, either polyphase or single-phase, are extensively used. For many applications, however, as especially for electrolytic work, direct currents are required, and are usually preferred also for electrical railroading and for low-tension distribution on the Edison three- wire system. Thus, where power is derived from an alternating system, transforming devices are required to convert from alternating to direct current. This can be done either by a direct-current generator driven by an alternating synchronous or induction motor, or by a single machine consuming alternating and pro- ducing direct current in one and the same armature. Such a machine is called a converter, and combines, to a certain extent, the features of a direct-current generator and an alternating synchronous motor, differing, however, from either in other features. Since in the converter the alternating and the direct current are in the same armature conductors, their e.m.fs. stand in a definite relation to each other, which is such that in practically all cases step-down transformers are necessary to generate the required alternating voltage. Comparing thus the converter with the combination of syn- chronous or induction motor and direct-current generator, the converter requires step-down transformers; the synchronous motor, if the alternating line voltage is considerably above 10,000 volts, generally requires step-down transformers also; with voltages of 1000 to 10,000 volts, however, , usually the synchronous motor and frequently the induction motor can be wound directly for the line voltage and stationary transformers saved. Thus on the one side we have two machines with or sometimes without stationarytransformers, on the other side a single machine with transformers. Regarding the reliability of operation and first cost, obviously a single machine is preferable. 223 224 ELEMENTS OF ELECTRICAL ENGINEERING Regarding efficiency, it is sufficient to compare the converter with the synchronous-motor-direct-current-generator set, since the induction motor is usually less efficient than the syn- chronous motor. The efficiency of stationary transformers of large size varies from 97 per cent, to 98 per cent., with an average of 97.5 per cent. That of converters or of synchronous motors varies between 91 per cent, and 95 per cent., with 93 per cent, as average, and that of the direct-current generator between 90 per cent, and 94 per cent., with 92 per cent, as average. Thus the converter with its step-down transformers will give an average efficiency of 90.7 per cent., a direct-current generator driven by synchronous motor with step-down transformers an efficiency of 83.4 per cent., without step-down transformers an efficiency of 85.6 per cent. Hence the converter is more efficient, and there- fore is almost always preferred. Mechanically the converter has the advantage that no transfer of mechanical energy takes place, since the torque consumed by the generation of the direct current and the torque produced by the alternating current are applied at the same armature con- ductors, while in a direct-current generator driven by a syn- chronous motor the power has to be transmitted mechanically through the shaft. EC. Ratio of e.m.fs. and of Currents 83. In its structure the synchronous converter consists of a closed-circuit armature, revolving in a direct-current excited field, and connected to a segmental commutator as well as to collector rings. Structurally it thus differs from a direct- current machine by the addition of the collector rings, from certain (now very little used) forms of synchronous machines by the addition of the segmental commutator. In consequence hereof, regarding types of armature windings and of field windings, etc., the same rule applies to the converter as to all commutating machines, except that in the converter the total number of armature coils with a series-wound armature, and the number of armature coils per pair of poles with a multiple- wound armature, must be divisible by the number of phases, and that multiple spiral and reentrant windings are difficult to apply. Regarding the wave shape of the alternating counter-gener- SYNCHRONOUS CONVERTERS 225 ated e.m.f., similar considerations apply as for a synchronous machine with closed-circuit armature; that is, the generated e.m.f. usually approximates a sine wave, due to the multi-tooth distributed winding. Thus, in the following, only those features will be discussed in which the synchronous converter differs from the commu- tating machines and synchronous machines treated in the preceding chapters. Fig. 122 represents diagrammatically the commutator of a direct-current machine with the armature coils A connected to adjacent commutator bars. The brushes are BiB 2 , and the field poles FiF 2 . If now two oppositely located points a ia 2 of the commutator are connected with two collector rings DiD 2 , it is obvious that '8 FIG. 122. Single-phase converter commutator. the e.m.f. between these points aia 2 , and thus between the collector rings DiZ> 2 , will be a maximum in the moment when the points aia 2 coincide with the brushes BiB 2 , and is in this moment equal to the direct voltage E of this machine. While the points ai 3 and D 4 an alter- nating voltage of the same frequency and intensity will be produced as between DI and D 2 , but in quadrature therewith, since at the moment where a 3 and a 4 coincide with the brushes BiB 2 and thus receive the maximum difference of potential, ai and az are at zero points of potential. Thus connecting four equidistant points a\, a 2 , 0,3, a 4 of the SYNCHRONOUS CONVERTERS 227 direct-current generator to four collector rings D\, D 2 , D 3 , D 4 , gives a four-phase converter of the e.m.f. EI = = E per phase. v 2 The current per phase is (neglecting losses and phase displace- ment) since the alternating power, 2 EJi, must equal the direct-current power, EI. Connecting three equidistant points of the commutator to three collector rings as in Fig. 124 gives a three-phase converter. 85. In Fig. 125 the three e.m.fs. between the three collector rings and the neutral point of the three-phase system (or Y voltages) are represented by the vectors OEi, OE Z , OEs, thus FIQ. 124. Three-phase syn- chronous converter. FIQ. 125. E.m.f. diagram of three-phase converter. the e.m.f. between the collector rings or the delta voltages by vectors EiE 2 , E 2 E 3 , and E$E\. The e.m.f. OEi is, however, nothing but half the e.m.f. EI in Fig. 122, of the single-phase Tjl converter, that is, = - = - Hence the Y voltage, or voltage 2 v 2 between collector ring and neutral point or center of the three- phase voltage triangle, is -= = 0.354 E. 2V2 and thus the delta voltage is E' = E l V3 0.612 E. 228 ELEMENTS OF ELECTRICAL ENGINEERING Since the total three-phase power 3 IiEi equals the total continuous-current power IE, it is In general, in an n-phase converter, or converter in which n equidistant points of the commutator (in a bipolar machine, or n equidistant points per pair of poles in a multipolar machine with multiple-wound armature) are connected to n collector rings, the voltage between any collector ring and the common neutral, or star voltage, is consequently the voltage between two adjacent collector rings, or ring voltage, is s' E sin- V2 since is the angular displacement between two adjacent col- lector rings. Herefrom the current per line, or star current, is found as 2V27 and the current from line to line, or from collector ring to ad- jacent collector ring, or ring current, is V2/ r = . 7T n sin n 86. As seen in the preceding, in the single-phase converter consisting of a closed-circuit armature tapped at two equi- distant points to the two collector rings, the alternating voltage is -= times the direct-current voltage, and the alternating cur- V 2 _ rent \/2 times the direct current. While such an arrangement of the single-phase converter is the simplest, requiring only two collector rings, it is undesirable, especially for larger machines, on account of the great total and especially local 7V heating in the armature conductors, as will be shown in the following, and SYNCHRONOUS CONVERTERS 229 due to the waste of e.m.f., since in the circuit from collector ring to collector ring the e.m.fs. generated in the coils next to the leads are wholly or almost wholly opposite to each other. The arrangement which I have called the two-circuit single- phase converter, and which is diagrammatically shown in Fig. 126, is therefore preferable. The step-down transformer T contains two independent secondary coils A and B, of which one, A y feeds into the armature over conductor rings DiD 2 and leads dia 2 , the other, B, over collector rings D 3 Z> 4 and leads a 3 a 4 , so that the two circuits aia z and a 3 a 4 are in phase with each other, and each spreads over 120 deg. arc instead of 180 deg. arc as in the single-circuit single-phase converter. a* FIG. 126. Two-circuit single-phase converter. In consequence thereof, in the two-circuit single-phase con- verter the alternating counter-generated e.m.f. bears to the con- tinuous-current e.m.f. the same relation as in the three-phase converter, that is, and from the equality of alternating- and direct-current power, 2 /i#i = IE, it follows that each of the two single-phase supply currents is -v/2 I f = -I = 0.8177. It is seen that in this arrangement one-third of the armature, from ai to a 3 and from a 2 to a 4 , carries the direct current only, the other two-thirds, from ai to a 2 andfroma 3 to a 4 , the differential current. x5 230 ELEMENTS OF ELECTRICAL ENGINEERING A six-phase converter is usually fed from a three-phase system by three transformers or one three-phase transformer. These transformers can either have each one secondary coil only of E twice the star or 7 voltage, = T=> which connects with its two terminals two collector rings leading to two opposite points of the armature, or each of the step-down transformers contains two independent secondary coils, and each of the two sets of secondary coils is connected in three-phase delta or F, but the one set of coils reversed with regard to each other, thus giving two three-phase systems which join to a six-phase system. The different transformer connections then are distinguished as "diametrical/ 7 "double delta" and "double F." For further arrangements of six-phase transformation, see "Theory and Calculation of Alternating-current Phenomena/ 7 fourth edition, Chapter XXXVI. The table below gives, with the direct-current voltage and direct current as unit, the alternating voltages and currents of the different converters. -u ** 11 11 1! V 43 A i 1 f 1 A dve-phase 1 " w fl.S 2 ,rj i a 08 hj H 00 H * i 1 i 1 i 1 1 i Volts between collector 2\/2 2V2 2V2 2-S/2 2V2 2V2 2\/2 ring and neutral point. = . 354 = 0.354 = . 354 = 0.354 = 0.354 = . 354 = . 354 1 V3 V3 1 . Tf sin Volts between adjacent collector rings 1.0 V2 = 0.707 2V2 = 0.612 2V2 = 0.612 ^=0.5 2V2 = 0.354 0.183 n vl Amperes per line V2 2-S/2 1 V2 2"V^2 V2 V3 3 V2 3 n 1.0 = 1.414 = 0.817 = 0.943 = . 707 = 0.472 0.236 Amperes between ad- jacent lines V2 V2 V3 2V 2 3V3 V2 3 . n . IT = 1.414 = 0.817 = 0.545 ^ = 0.5 = 0.472 0.455 n These currents give only the power component of alternating current corresponding to the direct-current output. Added thereto is the current required to supply the losses in the machine, that is, to rotate it, and the wattless component if a phase dis- placement is produced in the converter. SYNCHRONOUS CONVERTERS 231 HI. Variation of the Ratio of Electromotive Forces 87. The preceding ratios of e.m.fs. apply strictly only to the generated e.m.fs. and that under the assumption of a sine wave of alternating generated e.m.f. The latter is usually a sufficiently close approximation, since the armature of the converter is a multi-tooth structure, that is, contains a distributed winding. The ratio between the difference of potential at the commu- tator brushes and that at the collector rings of the converter usually differs somewhat from the theoretical ratio, due to the e.m.f. consumed in the converter armature, and in machines converting from alternating to continuous current, also due to the shape of the impressed wave. When converting from alternating to direct current, under load the difference of potential at the commutator brushes is less than the generated direct e.m.f., and the counter-generated alternating e.m.f. less than the impressed, due to the voltage consumed by the armature resistance. If the current in the converter is in phase with the impressed e.m.f., armature self-inductance has little effect, but reduces the counter-generated alternating e.m.f. below the impressed with a lagging and raises it with a leading current, in the same way as in a synchronous motor. Thus in general the ratio of voltages varies somewhat with the load and with the phase -relation, and with constant impressed alternating e.m.f. the difference of potential at the commutator brushes decreases with increasing load, decreases with decreasing excitation (lag), and increases with increasing excitation (lead). When converting from direct to alternating current the reverse is the case. The direct-current voltage stands in definite proportion only to the maximum value of the alternating voltage (being equal to twice the maximum star voltage), but to the effective value (or value read by voltmeter) only in so far as the latter depends upon the former, being = - 7= maximum value with a sine wave. Thus with an impressed wave of e.m.f. giving a different ratio of maximum to effective value, the ratio between direct and alternating voltage is changed in the same proportion as the ratio of maximum to effective; thus, for instance, with a flat-topped 232 ELEMENTS OF ELECTRICAL ENGINEERING wave of impressed e.m.f., the maximum value of alternating impressed e.m.f., and thus the direct voltage depending there- upon, are lower than with a sine wave of the same effective value, while with a peaked wave of impressed e.m.f. they are higher, by as much as 10 per cent, in extreme cases. In determining the wave shape of impressed e.m.f. at the con- verter terminals, not only the wave of generator e.m.f., but also that of the converter counter e.m.f., may be instrumental. Thus, with a converter connected directly to a generating system of very large capacity, the impressed e.m.f. wave will be practically identical with the generator wave, while at the terminals of a converter connected to the generator over long lines with re- active coils or inductive regulators interposed, the wave of im- pressed e.m.f. may be so far modified by that of the counter e.m.f. of the converter as to resemble the latter much more than the generator wave, and thereby the ratio of conversion may be quite different from that corresponding to the generator wave. Furthermore, for instance, in three-phase converters fed by ring or delta connected transformers, the star e.m.f. at the con- verter terminals, which determines the direct voltage, may differ from the star e.m.f. impressed by the generator, by con- taining different third and ninth harmonics, which cancel when compounding the star voltages to the delta voltage, and give identical delta voltages, as required. Hence, the ratios of e.m.fs. given in Section II have to be corrected by the drop of voltage in the armature, and have to be multiplied by a factor which is \/2 times the ratio of effective to maximum value of impressed wave of star e.m.f. (\/2 being the ratio of maximum to effective of the sine wave on which the ratios in Section II were based), that is, by a "form factor" of the e.m.f. wave. With an impressed wave differing from the sine shape, there is a current of higher frequency, but generally of negligible mag- nitude, through the converter armature, due to the difference between impressed and counter e.m.f. wave. IV. Armature Current and Heating 88. The current in the armature conductors of a converter is the difference between the alternating-current input and the direct-current output. SYNCHRONOUS CONVERTERS 233 In Fig. 127, ai, a 2 are two adjacent leads connected with the collector rings DI, D 2 in an n-phase converter. The alternating e.m.f. between a\ and a 2 , and thus the power component of the alternating current in the armature section between a\ and a 2 , will reach a maximum when this section is midway between the brushes BI and B z , as shown in Fig. 127. The direct current in every armature coil reverses at the mo- ment when the coil passes under brush BI or B 2 , and is thus a rec- tangular alternating current as shown in Fig. 128 as 7. At the moment when the power com- ponent of the alternating current is a maximum, an armature coil d midway between two adjacent alternating leads ai and a 2 is midway between the brushes BI and B 2} as in Fig. 127, and is thus in the middle of its rectan- gular continuous-current wave, and consequently in this coil the power component of the alternating current and the rectan- gular direct current are in phase with each other, but opposite, as FIG. 127. Diagram for study of armature heating in synchronous converters. FIG. 128. Direct current and alternating current in armature coil d, Fig. 127. FIG. 129. Resultant current in coil d, Fig. 127. shown in Fig. 128 as 7i and /, and the actual current is their difference, as shown in Fig. 129. In successive armature coils the direct current reverses suc- cessively; that is, the rectangular currents in successive arma- 234 ELEMENTS OF ELECTRICAL ENGINEERING 7 FIG. 130. Alternating current and direct current in coil between d and a\ or a* Fig. 127. FIG. 131. Resultant of currents given in Fig. 130. FIG. 132. Alternating current and direct current in coil between d and or a 2 , Fig. 127. FIG. 133. Resultant of currents shown in 132. SYNCHRONOUS CONVERTERS 235 ture coils are successively displaced in phase from each other; and since the alternating current is the same in the whole section ai a 2 , and in phase with the rectangular current in the coil d, it becomes more and more out of phase with the rectangular current when passing from coil d toward ai or a 2 , as shown in Figs. 130 to 133, until the maximum phase displacement between alternating and rectangular current is reached at the alternating leads ai and a 2 , and is equal to - li 89. Thus, if E = direct voltage, and I = direct current, in an armature coil displaced by angle T from the position d, mid- way between two adjacent leads of the n-phase converter, the direct current is ~ for the half period from to ?r, and the alter- nating current is V2 I' sin (0 - r), where /' = n sin - n is the effective value of the alternating current. Thus, the actual current in this armature coil is io = \/2 /' sin (0 T) - g _ / [4 sin (0-r) _ "2 nsin- ( n In a double-current generator, instead of the minus sign, a plus sign would connect the alternating and the direct current in the parenthesis. The effective value of the resultant converter current thus is: n sin - rnr sin - n n Since ~ is the current in the armature coil of a direct-current 236 ELEMENTS OF ELECTRICAL ENGINEERING generator of the same output, we have 7r = Jo' 2 o | i 16 COS*T 7 2 1 -1 n 2 sin 2 - n nir sin ^ 71 the ratio of the power loss in the armature coil resistance of the converter to that of the direct-current generator of the same output, and thus the ratio of coil heating. This ratio is a maximum at the position of the alternating leads, T = -, and is 7m = n* sin n It is a minimum for a coil midway between adjacent alter- nating leads, T = 0, and is = 8 ... i . 7T .IT n 2 sin 2 - mr sin - n n Integrating over T from (coil d) to-, that is, over the whole phase or section 0,1 0,%, we have the ratio of the total power loss in the armature resistance of an n-phase converter to that of the same machine as direct- current generator at the same output, or the relative armature heating. Thus, to get the same loss in the armature conductors, and consequently the same heating of the armature, the current in the converter, and thus its output, can be increased in the pro- portion 7= over that of the direct-current generator. The calculation for the two-circuit single-phase converter is somewhat different, since in this in one-third of the armature the Pr loss is that of the direct-current output, and only in the 27f other two-thirds or an arc -^ is there alternating current. o SYNCHRONOUS CONVERTERS 237 Thus in an armature coil displaced by angle r from the center of this latter section the resultant current is io = V2 /' sin (0 - r) - giving the effective value I III 16 W:= thus, the relative heating is //oV 11 16 ^ (I = \2/ with the minimum value at r = 0, it is " - T ~ = - 70 ' and with the maximum value at r = ^ it is o 11 8 =2 - 18; the average current heating in two-thirds of the armature is 11 48 TT T d T = -3- - ^/^ Sln 3 3 7T 2 in the remaining third of the armature, Tz = 1, thus the average is 3 = 1.151, and therefore the rating is -4= = 0.93. Vr By substituting for n, in the general equations of current heat- ing and rating based thereon, numerical values, we get the following table: 238 ELEMENTS OF ELECTRICAL ENGINEERING d V si 0> 0) Type Direct-cur generator P jj-fl a H a 1 1 8 n 2 2 3 4 6 12 1.00 0.45 0.70 0.225 0.20 0.19 0.187 7m r 1.00 1.00 3.00 1.37 2.18 1.157 1.20 0.555 0.73 0.37 0.42 0.26 0.24 0.20 0.187 Rating (by mean arm. heating) 1.00 0.85 0.93 1.34 1.64 1.96 2.24 2.31 As seen, in the two-circuit single-phase converter the arma- ture heating is less, and more uniformly distributed, than in the single-circuit single-phase converter. 90. A very great gain is made in the output by changing from three-phase to six-phase, but relatively little by still further increasing the number of phases. In these values, the small power component of current supply- ing the losses in the converter has been neglected. These values apply only to the case where the alternating current is in phase with the supply voltage, that is, for unity power-factor of supply. If, however, the current lags, or leads, by the time angle 0, then the alternating current and direct current are not in opposition in the armature coil d midway between adjacent leads, Fig. 127, and the resultant current is a minimum and of the shape shown in Fig. 128, at a point of the armature winding displaced from mid position d by angle r = 0. At the leads the displacement between alternating cur- 7T 7T rent and direct current then is not -, but - + 8 at the n n one, 6 at the other lead, and thus at the other side of the same n lead. The resultant current is thus increased at the one, de- creased at the other lead, and the heating changed accordingly. For instance, in a quarter-phase converter at zero phase dis- placement, the resultant current at the lead would be as shown in Fig. 134, - = 45 deg., while at 30 deg. lag the resultant currents in the two coils adjacent to the commutator lead are displaced SYNCHRONOUS CONVERTERS 239 respectively by- + & = 75 deg. and by - d = 15 deg., and so of very different shape, as shown by Figs. 135 and 136, giving very different local heating. Phase displacement thus increases the heating at the one, decreases it at the other side of each commutator lead. Let again, I = direct current per commutator brush. The effective value of the alternating power current in the armature winding, or ring current, corresponding thereto, is n sn - n Let pi' = total power current, 'allowing for the losses of power in the converter; ql f = reactive current in the converter, assumed as positive when lagging, as negative when leading, and si' = total current, where s = Vp 2 + tf 2 is the ratio of total current to the load current, that is, power current corresponding to the direct-current output, and = tan 6 is the time lag of the supply current; p is a quantity slightly larger than 1, by the losses in the converter, or slightly smaller than 1 in an inverted converter. The actual current in an armature coil displaced in position by angle r from the middle position d between the adjacent collector leads, then, is to = V2 I f [p sin (0 - T) - q cos (0 - r) } - ^ 4 [p sin (0 T) q cos (0 T)] I 2 | n sin - n and, therefore, its effective value is = \l I * * v jo / I 8(p 2 + g 2 ) 16 (p COST + gsinr) n / 8 s 2 16 s cos (T - 6) 2 1 1 + ~ rr^r ^r n 2 sin 2 - TTH sin - n n 240 ELEMENTS OF ELECTRICAL ENGINEERING \ FIG. 134. Quarter-phase converter unity power-factor, armature current at collector lead. \ \ v_ FIG. 135. Quarter-phase converter phase displacement 30 ture current at collector lead. 7 FIG. 136. Quarter-phase converter phase displacement 30 degrees, arma- ture current at collector lead. SYNCHRONOUS CONVERTERS 241 and herefrom the relative heating in an armature coil displaced by angle r from the middle between adjacent commutator leads: 8s 2 16scos(r-0) n sn - irn sn n n this gives at the leads, or for r = + > 8s n 2 sin 2 TTU sin n n 16 s cos ( 0) '8s 2 \n I n 2 sin 2 - irn sin - n n Averaging from to H gives the mean current-heating of Ti ft the converter armature. r 1 + J+^ %* -? n 2 sin 2 n 2 sin - n n i i 8s 2 16s cos i -f- 9 *7T 7T" 2 n 2 sin 2 n -t-i_ 8 (p 2 + q* ) 16 P 2 2 - n 91. This gives for Three-phase, n = 3: 7 T = 1 + 1.185 s 2 - 1.955 s cos (T - 0), y m = 1 + 1.185 s 2 - 1.955 s cos (60 0), r = 1 + 1.185s 2 - 1.620 p. Quarter-phase, n = 4: >Y T = 1 + s 2 - 1.795 s cos (r - 0), y m = 1 + s 2 - 1.795 s cos (45 0), T = 1 + s 2 - 1.620 . 242 ELEMENTS OF ELECTRICAL ENGINEERING Six-phase, n = 6: 7r = 1 + 0.889 s 2 - 1.695 s cos (r - 0), 7 m = 1 + 0.889 s 2 - 1.695 s cos (30 + 0), r = 1 + 0.889s 2 - 1.62 p, oo -phase, n = co : TT = 7m = r = 1 + 0.810 s 2 - 1.62 s cos = 1 + 0.810s 2 - 1.62 p. Choosing p = 1.04, that is, assuming 4 per cent, loss in friction and windage, core loss and field excitation the z' 2 r loss of the armature is not included in p, as it is represented by a drop of direct-current voltage below that corresponding to the alternat- ing voltage, and not by an increase of the alternating current over that corresponding to the direct current we get, for dif- ferent phase angles from = deg. to = 60 deg., the values given below: 0=0 10 20 30 40 50 60 s = -^ =1.04 1.0561.1081.20 1.36 1.62 2.08 cos q = s sin = react ' cur> =. 0.184 0.379 0.60 0.876 1.24 1.80 power cur. tan0 =0 0.176 0.364 0.577 0.839 1.192 1.732 Three-phase: _ i , 1.62 2 08 2.70 3. 65 5. 19 8.16 '* = *: 26 1.00 80 0.68 0. 70 0. 99 2.06 r = 0. 60 0.64 77 1.02 1. 51 2. 43 4.45 Quarter-phase : 7m = 1.02 1 39 1.88 2. 64 3. '87 6.30 7ft 7m' = 4 \J 0.55 43 0.38 0. 42 0. 73 1.71 r = - - 0. 40 0.43 54 0.75 1. 16 1. 94 3.64 Six-phase : 7m = \ o 44 0.62 88 1.27 1. 86 2. 85 4.85 7m' - J 1 0.31 24 0.25 0. 38 0. 75 1.79 r = 0. 28 0.31 41 0.60 0. 97 1. 65 3.17 oo -phase: -v 'v ' r Jm jm ~ * = 20 0.22 32 0.49 0. 82 1. 45 2.82 92. The values are shown graphically in Figs. 137 and 138, SYNCHRONOUS CONVERTERS 243 reactive current , . with tan 6 = - TT~ as abscissas, and 7 as ordinates energy current in Fig. 137, T as ordinates in Fig. 138. As seen, with increasing phase displacement, irrespectively whether lag or lead, the average as well as the maximum arma- ture heating very greatly increases. This shows the necessity of keeping the power-factor near unity at full load and overload, and when applied to phase control of the voltage by converter, means that the shunt field of the converter should be adjusted so as to give a considerable lagging current afho load, so that the ~7 7 I 7 GENERATOR HEA FIG. 137. Maximum 7 2 r heating in converter armature coil expressed in per cent, of direct-current generator 7 2 r heating. current comes into phase with the voltage at about full load. It therefore is very objectionable in this case to adjust the con- verter for minimum current at no load, as occasionally done by ignorant engineers, since such wrong adjustment would give con- siderable leading current at load, and therewith unnecessary armature heating. It must be considered, however, that above values are referred to the direct-current output, and with increase of phase angle the alternating-current input, at the same output, increases, 244 ELEMENTS OF ELECTRICAL ENGINEERING and the heating increases with the square of the current. Thus at 60 deg. lag or lead, the power-factor is 0.5, and the alternating- current input thus twice as great as at unity power-factor, corre- sponding to four times the heating. It is interesting therefore to refer the armature heating to the alternating-current input, that is, compare the heating of the converter with that of a synchronous motor of the same alternating-current input. This is given by r r 1 1 ~S PER CENT 60 / z / 320 / / 300/ / /, / / 260^ / / / ^ / / / / V / / / 200 <% f // / / / / 180 A 7 3\ &< / / 160 / ^ '* y / 140 / i / / r & 130 D REG" CU iR Ef T / / / / // \ 100 CEf ERA! OR HEAT ING, /^ / s / /" " 80 . ' ^ X x 60 - ^ ^ x^ ^ 40 , ' ^ ^^ F EAC 1 nvF CUR RFN- 1 PF CF^ T ?n 1 2 D 3 ) 4 OF 5 FUUL UOAD POWER CU.RRE 6,0 If) 80 90 100 1 NT 1 1 1 1 o FIG. 138. Average I z r heating in converter armature expressed in per cent. of direct-current generator 7 2 r heating. and, for p = 1.04; gives the following values: 0= ^ 10 20 30 40 50 60 tan (9 = 0.176 0.364 0.577 0.839 1.192 14.32 Three-phase: Quarter-phase : 0.5550.57 0.63 0.71 0.82 0.93 1.03 0.37 0.385 0.44 0.52 0.63 0.74 0.84 SYNCHRONOUS CONVERTERS 245 Six-phase: Ti = oo -phase: 0.26 0.28 0.335 0.42 0.52 0.63 0.73 0.185 0.197 0.26 0.34 0.44 0.55 0.65 It is seen that, compared with the total alternating-current input, the armature heating increases much less with increasing phase displacement, and is almost always much lower than the heating of the same machine at the same input and phase angle, when running a synchronous motor, as shown in Fig. 139. FIG. 139. Average I*r heating in converter armature expressed in per cent, of synchronous motor 7 2 r heating at the same power-factor. V. Armature Reaction 93. The armature reaction of the polyphase converter is the resultant of the armature reactions of the machine as direct- current generator and as synchronous motor. If the com- mutator brushes are set at right angles to the field poles or without lead or lag, as is usually done in converters, the direct- current armature reaction consists in a polarization in quadra- ture behind the field magnetism. The armature reaction due to the power component of the alternating current in a synchro- nous motor consists of a polarization in quadrature ahead of the field magnetism, which is opposite to the armature reaction as direct-current generator. Let m = total number of turns on the bipolar armature or per pair of poles of an n-phase converter, / = direct current, then the number of turns in series between the brushes = -~, hence the total armature ampere-turns, or polarization, = -^ Since, how- 16 * 246 ELEMENTS OF ELECTRICAL ENGINEERING ever, these ampere-turns are not unidirectional, but distributed over the whole surface of the armature, their resultant is ml avg. cos and, since avg. cos T 2 ml we have F = - = direct-current polarization of the converter 7T (or direct-current generator) armature. vn In an n-phase converter the number of turns per phase = n The current per phase, or current between two adjacent leads (ring current), is 7T n sin n hence, the ampere-turn per phase, ml' \/2 ml n n sin - n These ampere-turns are distributed over - of the circumference of the armature, and their resultant is thus ml' and, since we have n avg. cos avg. cos n . TT = - sin -i . ;' n irn = resultant polarization, in effective ampere-turns of one phase of the converter. The resultant m.m.f. of n equal m.m.fs. of effective value of FI, thus maximum value of FI \/2, acting under equal angles SYNCHRONOUS CONVERTERS 247 , and displaced in phase from each other by - of a period, or 71 71 phase angle , is found thus: = Fi\/2sin (0 -- j = one of the m.m.fs. of phase 2iir angle = -- , where i = 0, 1, 2 . . . n 1, acting in the direc- tion T = - ; that is. the zero point of one of the m.m.fs. FI is n ' taken as zero point of time 0, and the direction of this m.m.f. as zero point of direction T. The resultant m.m.f. in any direction r is thus and, since 2 iir 2 ir\ I 2 ir - )COB(T- we have that is, the resultant m.m.f. in any direction T has the phase 6 = r, and the intensity, rcFiA/2 ~^~ thus revolves in space with uniform velocity and constant in- tensity, in synchronism with the frequency of the alternating current. 248 ELEMENTS OF ELECTRICAL ENGINEERING Since in the converter, Fl = ^M, TTU we have the resultant m.m.f. of the power component of the alternating current in the n-phase converter. This m.m.f. revolves synchronously in the armature of the converter; and since the armature rotates at synchronism, the resultant m.m.f. stands still in space, or, with regard to the field poles, in opposition to the direct-current polarization. Since it is equal thereto, it follows that the resultant armature reac- tions of the direct current and of the corresponding power component of the alternating current in the synchronous con- verter are equal and opposite, thus neutralize each other, and the resultant armature polarization equals zero. The same is obviously the case in an inverted converter, that is, a machine changing from direct to alternating current. 94. The conditions in a single-phase converter are different, however. At the moment when the alternating current = 0, the full direct-current reaction exists. At the moment when the alternating current is a maximum, the reaction is the differ- ence between that of the alternating and of the direct current; and since the maximum alternating current in the single-phase converter equals twice the direct current, at this moment the resultant armature reaction is equal but opposite to the direct- current reaction. Hence, the armature reaction oscillates with twice the fre- quency of the alternating current, and with full intensity, and since it is in quadrature with the field excitation, tends to shift the magnetic flux rapidly across the field poles, and thereby tends to cause sparking and power losses. This oscillating reaction is, however, reduced by the damping effect of the mag- netic field structure. It is somewhat less in the two-circuit single-phase converter. Since in consequence hereof the commutation of the single- phase converter is not as good as that of the polyphase con- verter, in the former usually voltage commutation has to be resorted to; that is, a commutating pole used, or the brushes shifted from the position midway between the field poles; and SYNCHRONOUS CONVERTERS 249 in the latter case the continuous-current ampere-turns inclosed by twice the angle of lead of the brushes act as a demagnetizing armature reaction, and require a corresponding increase of the field excitation under load. While the absence of armature reaction eliminates the need of a commutating pole to counteract the sparking due to the re- verse field of armature reaction, nevertheless, commutating poles are very often used in converters, to control the high self- induction of commutation, which economical design requires in such machines. Such commutating poles contain only the am- pere turn required to produce the commutating flux, thus less than in generators. 95. Since the resultant main armature reactions neutralize each other in the polyphase converter, there remain only 1. The armature reaction due to the small power component of current required to rotate the machine, that is, to cover the internal losses of power, which is in quadrature with the field excitation or distorting, but of negligible magnitude. 2. The armature reaction due to the wattless component of alternating current where such exists. 3. An effect of oscillating nature, which may be called a higher harmonic of armature reaction. The direct current, as rectangular alternating current in the armature, changes in phase from coil to coil, while the alternating current is the same in a whole section of the armature between adjacent leads. Thus while the resultant reactions neutralize, a local effect remains which in its relation to the magnetic field oscillates with a period equal to the time of motion of the armature through the angle between adjacent alternating leads; that is, double frequency in a single-phase converter (in which it is equal in magnitude to the direct-current reaction, and is the oscillating armature reaction discussed above), sextuple frequency in a three-phase converter, and quadruple frequency in a four- phase converter. The amplitude of this oscillation in a polyphase converter is small, arid its influence upon the magnetic field is usually neg- ligible, due to the damping effect of the field spools, which act like a short-circuited winding for an oscillation of magnetism. A polyphase converter on unbalanced circuit can be con- sidered as a combination of a balanced polyphase and a single- phase converter; and since even single-phase converters operate quite satisfactorily, the effect of unbalanced circuits on the 250 ELEMENTS OF ELECTRICAL ENGINEERING polyphase converter is comparatively small, within reasonable limits. Since the armature reaction of the direct current and of the alternating current in the converter neutralize each other, no change of field excitation is required in the converter with changes of load. Furthermore, while in a direct-current generator the arma- ture reaction at given field strength is limited by the distortion of the field caused thereby, this limitation does not exist in a converter; and a much greater armature reaction can be safely used in converters than in direct-current generators, the dis- tortion being absent in the former. The practical limit of overload capacity of a converter is usu- ally far higher than in a direct-current generator, since the arma- ture heating is relatively small, and since the distortion of field, which causes sparking on the commutator under overloads in a direct-current generator, is absent in a converter. The theoretical limit of overload that is, the overload at which the converter as synchronous motor drops out of step and comes to a standstill is usually far beyond reach at steady frequency and constant impressed alternating voltage, while on an alternating circuit of pulsating frequency or drooping voltage it obviously depends upon the amplitude and period of the pulsation of frequency or on the drop of voltage. VI. Reactive Currents and Compounding 96. Since the polarization due to the power component of the alternating current as synchronous motor is in quadrature ahead of the field magnetization, the polarization or magnetizing effect of the lagging component of alternating current is in phase, that of the leading component of alternating current in oppositon to the field magnetization; that is, in the converter no magnetic distortion exists, and no armature reaction at all if the current is in phase with the impressed e.m.f., while the- armature reaction is demagnetizing with a leading and mag- netizing with a lagging current. Thus if the alternating Current is lagging, the field excitation at the same impressed e.m.f. has to be lower, and if the alter- nating current is leading, the field excitation has to be higher, than required with the alternating current in phase with the SYNCHRONOUS CONVERTERS 251 e.m.f. Inversely, by raising the field excitation a leading current, or by lowering it a lagging current, can be produced in a converter (and in a synchronous motor). Since the alternating current can be made magnetizing or demagnetizing according to the field excitation, at constant impressed alternating voltage, the field excitation of the con- verter can be varied through a wide range without noticeably affecting the voltage at the commutator brushes; and in con- verters of high armature reaction and relatively weak field, full load and overload can be carried by the machine without any field excitation whatever, that is, by exciting the field by armature reaction by the lagging alternating current. Such converters without field excitation, or reaction converters, must always run with more or less lagging current, that is, give the same reaction on the line as induction motors, which, as known, are far more objectionable than synchronous motors in their reaction on the alternating system, and therefore they are no longer used. Conversely, however, at constant impressed alternating vol- tage the direct-current voltage of a converter cannot be varied by varying the field excitation (except by the very small amount due to the change of the ratio of conversion), but a change of field excitation merely produces wattless currents, lagging or magnetizing with a decrease, leading or demagnetizing with an increase of field excitation. Thus to vary the continuous- current voltage of a converter usually the impressed alternating voltage has to be varied. This can be done either by potential regulator or compensator, that is, transformers of variable ratio of transformation, or by a synchronous machine of the same number of poles as the converter, on the same shaft and con- nected in series ("synchronous booster") or by the effect of watt- less currents on self-inductance. The latter method is especially suited for converters, due to their ability of producing wattless currents by change of .field excitation. The e.m.f. of self -inductance lags 90 deg. behind the current; thus, if the current is lagging 90 deg. behind the impressed e.m.f., the e.m.f. of self-inductance is 180 deg. behind, or in opposition to, the impressed e.m.f., and thus reduces it. If the current is 90 deg. ahead of the e.m.f., the e.m.f. of self-inductance is in phase with the impressed e.m.f., thus adds itself thereto and raises it. Therefore, if self-inductance is inserted into the lines between converter and constant-potential generator, and a watt- I 252 ELEMENTS OF ELECTRICAL ENGINEERING less lagging current is produced by the converter by a decrease of its field excitation, the e.m.f. of self-inductance of this lagging current in the line lowers the alternating impressed voltage at the converter and thus its direct-current voltage; and if a watt- less leading current is produced by the converter by an increase of its field excitation, the e.rn.f. of self-inductance of this leading current raises the impressed alternating voltage at the converter and thus its direct-current voltage. 97. In this manner, by self-inductance in the lines leading to the converter, its voltage can be varied by a change of field excitation, or conversely its voltage maintained constant at constant generator voltage or even constant generator excita- tion, with increasing load and thus increasing resistance drop in the line; or the voltage can even be increased with increasing load, that is, the system over-compounded. The change of field excitation of the converter with changes of load can be made automatic by the combination of shunt and series field, and in this manner a converter can be compounded or even over-compounded similarly to a direct-current generator. While the effect is the same, the action, however, is different; and the compounding takes place not in the machine as with a direct-current generator, but in the alternating lines leading to the machine, in which self-inductance becomes essential. As the reactance of the transmission line is rarely sufficient to give phase control over a wide range without excessive reac- tive currents, it is customary, especially at 25 cycles, to insert reactive coils into the leads between the converter and its step- down transformers, in those cases in which automatic phase control by converter series fields is desired, as in power trans- mission for suburban and interurban railways, etc., or to specially design the step-down transformers for high internal reactance. Usually these reactive coils are designed to give at full-load current a reactance voltage equal to about 15 per cent, of the converter supply voltage, and therefore capable of taking care of about 10 per cent, line drop at good power-factors. VII. Variable Ratio Converters ("Split Pole" Converters) 98. With a sine wave of alternating voltage, and the com- mutator brushes set at the magnetic neutral, that is, at right angles to the resultant magnetic flux, the direct voltage of a SYNCHRONOUS CONVERTERS 253 converter is constant at constant impressed alternating voltage. It equals the maximum value of the alternating voltage between two diametrically opposite points of the commutator, or "dia- metrical voltage," and the diametrical voltage is twice the voltage between alternating lead and neutral, or star or Y voltage of the polyphase system. A change of the direct voltage, at constant impressed alter- nating voltage, can be produced Either by changing the position angle between the commu- tator brushes and the resultant magnetic flux, so that the direct voltage between the brushes is not the maximum diametrical alternating voltage but only a part thereof, Or by changing the maximum diametrical alternating voltage, at constant effective impressed voltage, by wave-shape distortion by the superposition of higher harmonics. In the former case, only a reduction of the direct voltage below the normal value can be produced, while in the latter case an increase as well as a reduction can be produced, an increase if the higher harmonics are in phase, and a reduction if the higher harmonics are in opposition to the fundamental wave of the diametrical or Y voltage. Both methods are combined in the so-called " Regulating Pole Converter" or "Split Pole Converter," which is used to supply, from constant alternating voltage supply, direct voltage varying sometimes over a range of 20 per cent. In this type of converter, the field pole is divided into sections, usually two, a smaller one, the regulating pole, and a larger one, the main pole. By varying the excitation of the regulating pole from maximum in one direction, to maximum in the opposite direction, the direction of the resultant magnetic field flux, and the effective width of the field pole, and with the latter the wave shape, are varied. To keep the wave shape variation local in the converter, so as not to reflect it into the primary supply circuit, the proper transformer connection must be used. This is Y primary with preferably A or double delta (for three-phase and for six-phase) or Y and double Y or dia- metrical in the secondary. Vm. Starting 99. The polyphase converter is self-starting from rest; that is, when connected across the polyphase circuit it starts, acceler- 254 ELEMENTS OF ELECTRICAL ENGINEERING ates, and runs up to complete synchronism. The e.m.f. between the commutator brushes is alternating in starting, with the fre- quency of slip below synchronism. Thus a direct-current volt- meter or incandescent lamps connected across the commutator brushes indicate by their beats the approach of the converter to synchronism. When starting, the field circuit of the converter has to be opened or at least greatly weakened. The starting of the polyphase converter is largely a hysteresis effect and entirely so in machines with laminated field poles, while in ma- chines with solid magnet poles or with a short-circuited winding (squirrel-cage) in the field poles, secondary currents in the latter contribute to the starting torque, but at the same time reduce the magnetic starting flux by their demagnetizing effect. The torque is produced by the attraction between the alternating currents of the successive phases upon the remanent magnetism and secondary currents produced by the preceding phase. It is necessarily comparatively weak, and from full-load to twice full-load current at from one-third to one-half of full voltage is required to start from rest without load. Usually, low-voltage taps on the transformers are used to give the lower starting voltage. While an induction motor can never reach exact synchronism, but must even at no load slip slightly to produce the friction torque, the converter or synchronous motor reaches exact syn- chronism, due to the difference of the magnetic reluctance in the direction of the field poles and in the direction in electrical quadrature thereto; that is, the field structure acts like a shuttle armature and the polar projections catch with the rotating magnet poles in the armature, in a similar way as an induction motor armature with a single short-circuited coil (synchronous induction motor, reaction machine) drops into step. Obviously, the single-phase converter is not self-starting. At the moment of starting, the field circuit of the converter is in the position of a secondary to the armature circuit as primary; and since in general the number of field turns is very much larger than the number of armature turns, excessive e.m.fs. may be generated in the field circuit, reaching frequently 4000 to 6000 volts, which have to be taken care of by some means, as by breaking the field circuit into sections, or protecting against ex- cessive voltages by a squirrel-cage starting winding in the pole faces. As soon as synchronism is reached, which usually takes SYNCHRONOUS CONVERTERS 255 from a few seconds to a minute or more, and is seen by the ap- pearance of continuous voltage at the commutator brushes, the field circuit is closed and the load put on the converter. Ob- viously, while starting, the direct-current side of the converter must be open-circuited, since the e.m.f. between commutator brushes is alternating until synchronism is reached. When starting from the alternating side, the converter can drop into synchronism at either polarity; but its polarity can be reversed by strongly exciting the field in the right direction by some outside source, as another converter, etc., or by momen- tarily opening the circuit and thereby letting the converter slip one pole. Since when starting from the alternating side the converter requires a very large and, at the same time, lagging current, it is occasionally preferable to start it from the direct-current side as direct-current motor. This can be done when connected to storage battery or direct-current generator. When feeding into a direct-current system together with other converters or con- verter stations, all but the first converter can be started from the continuous current side by means of rheostats inserted into the armature circuit. To avoid the necessity of synchronizing the converter, by phase lamps, with the alternating system in case of starting by direct current (which operation may be difficult where the direct voltage fluctuates, owing to heavy fluctuations of load, as rail- way systems), it is frequently preferable to run the converter up to or beyond synchronism by direct current, then cut off from the direct current, open the field circuit and connect it to the alternating system, thus bringing it into step by alternating current. If starting from the alternating side is to be avoided, and direct current not always available, as when starting the first converter, a small induction motor (of less poles than the con- verter) is used as starting motor. Converters usually are started from the alternating side. IX. Inverted Converters 100. . Converters may be used to change either from alter- nating to direct current or as inverted converters from direct to alternating current. While the former use is by far the more 256 ELEMENTS OF ELECTRICAL ENGINEERING frequent, sometimes inverted converters are desirable. Thus in low-tension direct-current systems outlying districts have been supplied by converting from direct to alternating, transmitting as alternating, and then reconverting to direct current. Or in a station containing direct-current generators for short-distance supply and alternators for long-distance supply, the converter may be used as the connecting link to shift the load from the direct to the alternating generators, or inversely, and thus be operated either way according to the distribution of load on the system. Or inverted operation may be used in emergencies to produce alternating current. When converting from alternating to direct current, the speed of the converter is rigidly fixed by the frequency, and cannot be varied by its field excitation, the variation of the latter merely changing the phase relation of the alternating current. When converting, however, from direct to alternating current as the only source of alternating current, that is, not running in multiple with engine- or turbine-driven alternating-current generators, the speed of the converter as direct-current motor depends upon the field strength; thus it increases with decreasing and decreases with increasing field strength. As alternating-current generator, however, the field strength depends upon the intensity and phase relation of the alternating current, lagging current reducing the field strength and thus increasing speed and frequency, and leading current increasing the field strength and thus decreasing speed and frequency. Thus, if a load of lagging current is put on an inverted con- verter, as, for instance, by starting an induction motor or another converter thereby from the alternating side, the demagnetizing effect of the alternating current reduces the field strength and causes the converter to increase in speed and frequency. An in- crease of frequency, however, may increase the lag of the current, and thus its demagnetizing effect, and thereby still further in- crease the speed, so that the acceleration may become so rapid as to be beyond control by the field rheostat and endanger the machine. Hence inverted converters have to be carefully watched, especially when starting other converters from them, and some absolutely positive device is necessary to cut the in- verted converter off the circuit entirely as soon as its speed ex- ceeds the danger limit. The relatively safest arrangement is separate excitation of the inverted converter by an exciter SYNCHRONOUS CONVERTERS 257 mechanically driven thereby, since an increase of speed in- creases the exciter voltage at a still higher rate, and thereby the excitation of the converter, and thus tends to check its speed. This danger of racing does not exist if the inverted converter operates in parallel with alternating generators, provided that the latter and their prime movers are of such size that they cannot be carried away in speed by the converter. In an in- verted converter running in parallel with alternators the speed is not changed by the field excitation, but a change of the latter merely changes the phase relation of the alternating current supplied by the converter; that is, the converter receives power from the direct-current system, and supplies power into the alter- nating-current system but at the same time receives wattless current from the alternating system, lagging at under-excitation, leading at over-excitation, and can in the same way as an ordinary converter or synchronous motor be used to compensate for watt- less currents in other parts of the alternating system, or to regu- late the voltage by phase control. X. Frequency 101. While converters can be designed for any frequency, the use of high frequency, as 60 cycles, imposes more severe limita- tions on the design, especially that of the commutator, as to make the high-frequency converter inferior to the low-frequency or 25-cycle converter. The commutator surface moves the distance from brush to next brush, or the commutator pitch, during one-half cycle, that is, 3^50 second with a 25-cycle, J^20 second with a 60-cycle converter. The peripheral speed of the commutator, however, is limited by mechanical, electrical, and thermal considera- tions centrifugal forces, loss of power by brush friction, and heating caused thereby. The limitation of peripheral speed limits the commutator pitch. Within this pitch must be in- cluded as many commutator segments as necessary to take care of the voltage from brush to brush, and these segments must have a width sufficient for mechanical strength. With the smaller pitch required for high frequency, this may become impossible, and the limits of conservative design thus may have to be exceeded. In a converter, due to the absence of armature reaction and field distortion, a higher voltage per commutator segment can be 258 ELEMENTS OF ELECTRICAL ENGINEERING . allowed than in a direct-current generator. Assuming 17 volts as limit of conservative design would give for a 600-volt con- verter 36 segments from brush to brush. Allowing 0.2 inch for segment and insulation, as minimum conservative value, 37 segments give a pitch of 7.4 inches. Estimating 5000 feet per minute as conservative limit of commutator speed gives 83.3 feet or 1000 inches peripheral speed per second, and with 7.4 inches pitch this gives 136 half cycles, or 68 cycles, as limit of the frequency, permitting conservative commutator design. At 60 cycles higher voltage per segment, narrower segments and higher commutator speeds thus are necessary than at 25 cycles, and the 60-cycle converter, though still within conserva- tive limits, does not permit as conservative commutator design, especially at higher voltage, as a low-frequency converter, and a lower self-inductance of commutation thus must be aimed at than permissible in a 25-cycle converter, the more so as the fre- quency of commutation (half the number of commutator seg- ments per pole times frequency of rotation) necessarily is higher in the 60-cycle converter. . . Somewhat similar considerations also apply to the armature construction : the peripheral speed of the armature, even if chosen higher for the 60-cycle converter, limits the pitch per pole at the armature circumference, and thereby the ampere conductors per pole and thus the armature reaction, the more so as shallower slots are necessary. The 60-cycle converter cannot be built with anything like the same armature reaction as is feasible at lower frequency. On the armature reaction, however, very largely depends the stability of a synchronous motor or converter, and machines of low armature reaction tend far more to surging and pulsation of current and voltage than machines of high armature reaction. The 60-cycle converter therefore cannot be made quite as stable and capable of taking care of violent fluctuations of load and of excessive overloads as 25-cycle converters can, and in this respect the lower-frequency machine is preferable, though under reasonably favorable conditions regarding variations of load, variations of supply voltage, and overload 60-cycle con- verters give excellent service. It is this inherent inferiority of the 60-cycle converter which has largely been instrumental in introducing 25 cycles as the frequency of electric power generation and distribution. SYNCHRONOUS CONVERTERS 259 At 25 cycles, converters are used on railway load the most fluctuating and therefore most severe service built for 1200 volts, and even still much higher voltages are available. XI. Double-current Generators 102. Similar in appearance to the converter, which changes from alternating to direct current, and to the inverted converter, which changes from direct to alternating current, is the double- current generator; that is, a machine driven by mechanical power and producing direct current as well as alternating current from the same armature, which is connected to commutator and col- lector rings in the same way as in the converter. Obviously the use of the double-current generator is limited to those sizes and speeds at which a good direct-current generator can be built with the same number of poles as a good alternator, that is, low- frequency machines of large output and relatively high speed; while high-frequency low-speed double-current generators are undesirable. The essential difference between double-current generator and converter is, however, that in the former the direct current and the alternating current are not in opposition as in the latter, but in the same direction, and the resultant armature polarization thus the sum of the armature polarization of the direct current and of the alternating current. Since at the same output and the same field strength the arma- ture polarization of the direct current and that of the alternating current are the same, it follows that the resultant armature polari- zation of the double-current generator is proportional to the load regardless of the proportion in which this load is distributed between the alternating- and direct-current sides. The heating of the armature due to its resistance depends upon the sum of the two currents, that is, upon the total load on the machine. Hence, the output of the double-current generator is limited by the current heating of the armature and by the field distortion due to the armature reaction, in the same way as in a direct-current generator or alternator, and is consequently much less than that of a converter. In double-current generators, owing to the existence of arma- ture reaction and consequent field distortion, the commutator brushes are more or less shifted against the neutral, and the 260 ELEMENTS OF ELECTRICAL ENGINEERING direction of the continuous-current armature polarization is thus shifted against the neutral by the same angle as the brushes. The direction of the alternating-current armature polarization, however, is shifted against the neutral by the angle of phase displacement of the alternating current. In consequence thereof, the reactions upon the field of the two parts of the armature polari- zation, that due to the continuous current and that due to the alternating current, are usually different. The reaction on the field of the direct-current load can be overcome by a series field. The reaction on the field of the alternating-current load when feeding converters can be compensated for by a change of phase relation, by means of a series field on the converter, with self- inductance in the alternating lines, or reactive coils at the converters. Thus, a double-current generator feeding on the alternating side converters can be considered as a direct-current generator in which a part of the commutator, with a corresponding part of the series field, is separated from the generator and located at a distance, connected by alternating leads to the generator. Ob- viously, automatic compounding of a double-current generator is feasible only if the phase relation of the alternating current changes from lag at no load to lead at load, in the same way as produced by a compounded converter. Otherwise, rheostatic control of the generator is necessary. This is, for instance, the case if the voltage of the double-current generator has to be varied to suit the conditions of its direct-current load, and the voltage of the converter at the end of the alternating lines varied to suit the conditions of load at the receiving end, independent of the voltage at the double-current generator, by means of alternating "potential regulators or compensators. Compared with the direct-current generator, the field of the double-current generator must be such as to give a much greater stability of voltage, owing to the strong demagnetizing effect which may be exerted by lagging currents on the alternating side, and may cause the machine to lose its excitation altogether. For this reason it is frequently preferable to excite double-current generators separately. With the general adoption of large three-phase steam-turbine units for electric power generation, the use of inverted converter and double-current generator has greatly decreased. SYNCHRONOUS CONVERTERS 261 XII. Conclusion 103. Of the types of machines, converter, inverted converter, and double-current generator, sundry combinations can be de- devised with each other and with synchronous motors, alternators, direct-current motors and generators. Thus, for instance, a converter can be used to supply a certain amount of mechanical power as synchronous motor. In this case the alternating current is increased beyond the value corresponding to the direct current by the amount of current giving the mechanical power, and the armature reactions do not neutralize each other, but the reaction of the alternating current exceeds that of the direct current by the amount corresponding to the mechanical load. In the same way the current heating of the armature is in- creased. An inverted converter can also be used to supply some mechanical power. Either arrangement, however, while quite feasible, has the disadvantage of interfering with auto- matic control of voltage by compounding. Double-current generators can be used to supply more power into the alternating circuit than is given by their prime mover, by receiving power from the direct-current side. In this case a part of the alternating power is generated from mechanical power, and the other converted from direct-current power, and the machine combines the features of an alternator with those of an inverted converter. Conversely, when supplying direct-current power and receiving mechanical power from the prime mover and electric power from the alternating system, the double-current generator combines the features of a direct-current generator and a converter. In either case the armature reaction, etc., are the sum of those corresponding to the two types of machines combined. 104. A combination of the converter with the direct-current generator is represented by the so-called "motor converter," which consists of the concatenation of a commutating machine with an induction machine. If the secondary of an induction machine is connected to a second induction or synchronous machine on the same shaft, and of the same number of poles, the combination runs at half synchronous speed, and the first induction machine as frequency converter supplies half of its power as electric power of half frequency to the second machine, and changes the other half 262 ELEMENTS OF ELECTRICAL ENGINEERING as motor into mechanical power, driving the second machine as generator. (Or, if the two machines have different number of poles, or are connected to run at different speeds, the division of power is at a different but constant ratio) . Using thus a double- current generator as second machine, it receives half of its power mechanically, by the induction machine as motor, and the other half electrically, by the induction machine as frequency converter. Such a machine, then, is intermediate between a converter and a direct-current generator, having an armature reaction equal to half that of a direct-current generator. Such motor converters have been recommended for high-fre- quency systems, as their commutating component is of half frequency, and thus affords a better commutator design than a high-frequency converter. They are necessarily much larger than standard converters, but are smaller than motor generator sets, as half the power is converted in either machine. One advantage of this type of machine for phase control is that it requires no additional reactive coils, as the induction machine affords sufficient reactance. The use of the converter to change from alternating to alter- nating of a different phase, as, for instance, when using a quarter- phase converter to receive power by one pair of its collector rings from a single-phase circuit and supplying from its other pair of collector rings the other phase of a quarter-phase system, or a three-phase converter on a single-phase system supplying the third wire of a three-phase system from its third collector ring, . is outside the scope of this treatise, and is, moreover, of very little importance^ since induction or synchronous motors are superior in this respect. APPENDIX Xin. Direct-current Converter 105. If n equidistant pairs of diametrically opposite points of a commutating machine armature are connected to the ends of n compensators or autotransformers, that is, electric circuits interlinked with a magnetic circuit, and the centers of these auto- transformers connected with each other to a neutral point as shown diagrammatically in Fig. 140 for n = 3, this neutral is equidis- tant in potential from the two sets of commutator brushes, and such a machine can be used as continuous current converter, to SYNCHRONOUS CONVERTERS 263 transform in the ratio of potentials 1 :2 or 2 : 1 or 1 : 1, in the latter case transforming power from one side of a three- wire system to the other side. Obviously either the n autotransformers can be stationary and connected to the armature by 2 n collector rings, or the auto- transformers rotated with the armature and their common neutral connected to the external circuit by one collector ring. The distribution of potential and of current in such a direct- current converter is shown in Fig. 141 for n = 2, that is, two autotransformers in quadrature. With the voltage 2 e between the outside conductors of the FIG. 140. Diagram of direct-current converter. system, the voltage between the neutral and outside conductor is e, that on each of the 2 n autotransformer sections is e sin(0 J, k = 0, 1, 2 . . . 2 n 1. Neglecting losses in the converter and the autotransformer, the currents in the two sets of commutator brushes are equal and of the same direction, that is, both outgoing or both incoming, and opposite to the current in the neutral ; that is, two equal currents i enter the commutator brushes and issue as current 2 i from the neutral, or inversely. From the law of conservation of energy it follows that the cur- rent 2 i entering from the neutral divides in 2 n equal and constant branches of direct current, , in the 2 n autotransformer sections, ' n 1 and hence enters the armature, to issue as current i from each of the commutator brushes. 264 ELEMENTS OF ELECTRICAL ENGINEERING In reality the current in each autotransformer section is *7* / irJf \ -- h io \/2 cos ( e 60 ---- h ) t Ti \ Ti I where i Q is the exciting current of the magnetic circuit of the auto- transformer, and a the angle of hysteretic advance of phase. At the commutator the current on the motor side is larger than the current on the generator side, by the amount required to cover the losses of power in converter and autotransformer. In Fig. 141 the positive side of the system is generator, the negative side motor. This machine can be considered as receiv- ing the current i at the voltage e from the negative side of the system, and transforming it into current i at voltage e on the 21 FIG. 141. Distribution e.m.f . and current in direct-current converter. positive side of the system, or it can be considered as receiving current i at voltage 2e from the system, and transforming it into current 2 i at the voltage e on the positive side of the system, or of receiving current 2 i at voltage e from the negative side, and returning current i at voltage 2e. In either case the direct- current converter produces a difference of power of 2 ie between the two sides of the three- wire system. The armature reaction of the currents from the generator side of the converter is equal but opposite to the armature reaction of the corresponding currents entering the motor side, and the motor and generator armature reactions thus neutralize each other, as in the synchronous converter; that is, the resultant SYNCHRONOUS CONVERTERS 265 armature reaction of the continuous-current converter is prac- tically zero, or the only remaining armature reaction is that corresponding to the relatively small current required to rotate the machine, that is, to supply the internal losses in the same. The armature reaction of the current supplying the electric power transformed into mechanical power obviously also remains, if the machine is used simultaneously as motor, as for driving a booster connected into the system to produce a difference between the voltages of the two sides, or the armature reaction of the currents generated from mechanical power if the machine is driven as generator. I I. Ill jiHojifljybJU yyyyyyyyyyyyyyywyy jyyyyy a'"o"o a^a z a 3 C B J 1 a"a"a 1J. 2i FIG. 142. Development of a direct-current converter. 106. While the currents in the armature coils are more or less sine waves in the alternator, rectangular reversed currents in the direct-current generator or motor, and distorted triple-fre- quency currents in the synchronous converter, the currents in the armature coils of the direct-current converter are approximately triangular double-frequency waves. Let Fig. 142 represent a development of a direct-current con- verter with brushes BI and B 2 , and C one autotransformer re- ceiving current 2 i from the neutral. Consider first an armature coil ai adjacent and behind (in the direction of rotation) an auto- transformer lead 61. In the moment when autotransformer leads 61 6 2 coincide with the brushes BI B% the current i directly enters 266 ELEMENTS OF ELECTRICAL ENGINEERING the brushes and coil ai is without current. In the next moment (Fig. 142A) the total current i from 61 passes coil ai to brush Bi t while there is yet practically no current from 61 over coils a' a", etc., to brush B z . But with the forward motion of the arma- ture less and less of the current from 61 passes through i a 2 , etc., to brush BI and more over a' a", etc., to brush B^ until in the position of a\ midway between 61 and 6 2 (Fig. 1425), one-half of the current from 61 passes ai a 2 , etc., to BI, the other half a' a", etc., to B%. With the further rotation the current in a\ grows less and becomes zero when 61 coincides with B%, or half a cycle after its coincidence with BI. That is, the current in FIG. 143. Current in the various coils of a direct-current converter. coil ai approximately has the triangular form shown as ii in Fig. 143, changing twice per period from to i. It is shown negative, since it is against the direction of rotation of the armature. In the same way we see that the current in the coil a', adjacent ahead of the lead 61, has a shape shown as i' in Fig. 143. The current in coil a midway between two commutator leads has the form io, and in general the current in any armature coil a x , dis- tant by angle r from the midway position a , has the form i x , Fig. 143. All the currents become zero at the moment when the autotrans- former leads 61 6 2 coincide with the brushes BI B 2 , and change SYNCHRONOUS CONVERTERS 267 by i at the moment when their respective coils pass a commu- tator brush. Thus the lines A and A' in Fig. 144 with zero values at BI B^ the position of brushes, represent the currents in the individual armature coils. The current changes from A to A' at the moment = r when the respective armature coil passes the brush, twice per period. Due to the inductance of the armature coils, which opposes the change of current, the current waves are not perfectly triangular, but differ somewhat therefrom. With n autotransformers, each autotransformer lead carries the current , which passes through the armature coils as triangular n current, changing by in the moment the armature coil passes a commutator brush. This current passes the zero value in the moment the autotransformer lead coincides with a brush. Thus, FIG. 144. Current in individual coils of a direct-current converter with one compensator. the differents current of n autotransformers which are superposed in an armature coil a x have the shape shown in Fig. 199 for n = 3. That is, each autotransformer gives a set of slanting lines AiA'i, A^A'z, AsA's, and all the branch currents i\, iz, i z , super- posed, give a resultant current i x , which changes by i in the moment the coil passes the brush. i x varies between the extreme values (2 p 1) and ^(2p + 1), if the armature coil is dis- placed from the midway position between two adjacent autotrans- former leads by angle r, and p = - p varies between ^ and Thus the current in an armature coil in position p = - can 268 ELEMENTS OF ELECTRICAL ENGINEERING be denoted in the range from p to 1 + p, or r to TT + r, by where e x = - 7T \\ / "1?t A'f A", / A" 2 A FIG. 145. Current in a single coil of a direct-current converter with three compensators. The effective value of this current is / = Since in the same machine as direct-current generator at voltage 2 e and current i] the current per armature coil is ~> the ratio of current is I i 2 SYNCHRONOUS CONVERTERS 269 and thus the relative Pr loss or the heat developed in the armature coil, /A 2 *- i with a minimum, and a maximum, p = 0, TO = M> 1 ^ 1 = ' Jm 3^w 2 3w 2 The mean heating or Pr of the armature is found by integrating over 7 from 1 1 as r = n J_ 2n ydp 2n I_L JL^ _ 1 + n 2 3n 2 ' This gives the following table, for the direct-current converter, of minimum current heating, 70, in the coil midway between DIRECT-CURRENT CONVERTER IV RATING d. c. No. of compensators, n = gen. 1 2 3 4 n 00 Minimum current heating P = 0, 70 = 1 ' K K H H H H Maximum current heating, 1 i 1 % J^2 ^ i^g /^ n' n Mean current heating, 1 r = 1 H ^2 !%7 1%8 H Rating, 1 Vr" 1 1.225 1.549 1.643 1.681 / 3n 2 1.732 \l+n 2 270 ELEMENTS OF ELECTRICAL ENGINEERING adjacent commutator leads, maximum current heating, y m , in the coil adjacent to the commutator lead, mean current heating, r, and rating as based on mean current heating in the armature, 1 . As seen, the output of the direct-current converter is greater than that of the same machine as generator. Using more than three autotransformers offers very little advantage, and the dif- ference between three and two autotransformers is comparatively small, also, but the difference between two and one autotransfor- mer, especially regarding the local armature heating, is considera- ble, so that for most practical purposes a two-autotransformer converter would be preferable. The number of autotransformers used in the direct-current converter has a similar effect regarding current distribution, heating, etc., as the number of phases in the synchronous converter. Obviously these relative outputs given in above table refer to the armature heating only. Regarding commutation, the total current at the brushes is the same in the converter as in the generator, the only advantage of the former being the better commutation due to the absence of armature reaction. The limit of output set by armature reaction and correspond- ing field excitation in a motor or generator obviously does not exist at all in a converter. It follows herefrom that a direct- current motor or generator does not give the most advantageous direct-current converter, but that in the direct-current converter just as in the synchronous converter, it is preferable to propor- tion the parts differently in accordance with above discussion, as, for instance, to use less conductor section, a greater number of conductors in series per pole, etc. XIV. Three-wire Generator and Converter 107. A machine based upon the principle of the direct-current converter is frequently used to supply a three- wire direct-current distribution system (Edison system). This machine may be a single generator or synchronous converter, which is designed for the voltage between the outside conductors of the circuit (the positive and the negative conductor), 220 to 280 volts, while the middle conductor of the system, or neutral conductor, is con- SYNCHRONOUS CONVERTERS 271 nected to the generator by autotransformer and collector rings, or, in the case of a synchronous converter, is connected to the neutral of the step-up transformers, and the latter thus used as autotransformers. -*; to v 2 "* n C, 2 1 , o )T ' k g to t.< 2 FIG. 146. Three-wire machine with single autotransformer. A three-wire generator thus is a combination of a direct- current generator and a direct-current converter, and a three- wire converter is a combination of a synchronous converter and a direct-current converter. Such a three-wire machine has the advantage over two separate machines, connected to the two 2C i-l. < Fro. 147. Three-wire system with two machines. sides of the three- wire direct-current system, of combining two smaller machines into one of twice the size, and thus higher space- and operation-economy and lower cost, and has the further advantage that only half as large current is commutated as by 272 ELEMENTS OF ELECTRICAL ENGINEERING the use of two separate machines; that is, the positive brush of the machine on the 'negative and the negative brush of the machine on the positive side of the system are saved, as seen by the diagrammatic sketch of the machine in Fig. 146 and the two separate two-wire machines in Fig. 147. The use of three-wire 220-volt machines on three-wire direct-current systems thus has practically displaced that of two separate 110- volt machines. A. THREE-WIRE DIRECT-CURRENT GENERATOR 108. In such machines, either only one compensator or auto- transformer is used for deriving the neutral, as shown diagram- matically in Fig. 146, or two autotransformers in quadrature, as shown in Fig. 148, but rarely more. FIG. 148. Three-wire machine with two compensators. As the efficiency of conversion of a direct-current converter with two autotransformers in quadrature (Fig. 148) is higher than that of a direct-current converter with single autotransformer (Fig. 146), it is preferable to use two (or even more) autotrans- formers where a large amount of power is to be converted, that is, where a very great unbalancing between the two sides of the three-wire system may occur, or one side may be practically unloaded while the other is overloaded. Where, however, the load is fairly distributed between the two sides of the system, that is, the neutral current (which is the difference between the currents on the two sides of the system) is small and so only a small part of the generator power is converted from one side to the other, and the efficiency of this conversion thus of negligible SYNCHRONOUS CONVERTERS 273 influence on the heating and the output of the machine, a single autotransformer is preferable because of its simplicity. In three- wire distribution systems the latter is practically always the case, that is, the load fairly balanced and the neutral current small. The size of the autotransformers depends upon the amount of unbalanced power, that is, the maximum difference between the load on the two sides of the three- wire system, and thus equals the product of neutral current i Q and voltage e between neutral and outside conductor; that is, in the three- wire system of vol- tage e per circuit, voltage 2e between the outside conductors, and maximum current i in the outside conductors, the generator power rating is p = 2 ei. Let now io = maximum unbalanced current in the neutral usually not exceeding 10 to 20 per cent, of i and using a single autotransformer, connected diametrically across the armature, Fig. 146, the maximum of the alternating voltage which it re- ceives is 2 e, and its effective voltage therefore e \/2. As the neutral current i Q divides when entering the autotransformer, the current in the compensating winding is -^ (neglecting the small z exciting current), and the volt-ampere capacity of the autotrans- former thus is and PQ _ 1 io P ~ 2 V2 i = 0.354 *- x Even with the neutral current equal to the current in the out- side conductor, or the one side of the system fully loaded, the other not loaded, the autotransformer thus would have only 35.4 per cent, of the volt-ampere capacity of the generator, and as an autotransformer of ratio 1 -r- 1 is half the size of a trans- former of the same volt-ampere capacity, in this case the auto- transformer has, approximately, the size of a transformer of 17.7 per cent, of the size of the generator. With the maximum unbalancing of 20 per cent., or -r- = 0.2, 274 ELEMENTS OF ELECTRICAL ENGINEERING the autotransformer thus has 7 per cent, of the volt-ampere capacity of the generator, or the size of a transformer of only 3.5 per cent, of the generator capacity, that is, is very small, and this method is therefore the most convenient for deriving the neutral of a three- wire distribution system. When using n autotransformers, obviously each has - of the size which a single autotransformer would have. The disadvantage of the three-wire generator over two sepa- rate generators is that a three-wire generator can only divide the voltage in two equal parts, that is, the two sides of the system have the same voltage at the generator. The use of two separate generators, however, permits the production of a higher voltage on one side of the system than on the other, and thus takes care of the greater line drop on the more evenly loaded side. Even in the case, however, where a voltage difference between the two sides of the system is desired for controlling feeder drops, it can more economically be given by a separate booster in the neu- tral, as such a booster woul'd require only a capacity equal to the neutral current times half the desired voltage difference between the two sides, and with 20 per cent, neutral current and 10 per cent, voltage difference between the two sides, thus would have only 1 per cent, of the size of the generator. B. THBEE-WIRE CONVERTER 109. In a converter feeding a three-wire direct-current system the neutral can be derived by connection to the transformer neutral. Even in this case, however, frequently a separate auto- transformer is used, connected across a pair of collector rings of the converter, since, as seen above, with the moderate unbalanc- ing usually existing, such a compensator is very small. When connecting the direct-current neutral to the transformer neutral it is necessary to use such a connection that the trans- former can operate as autotransformer, that is, that the direct current in each transformer divides into two branches of equal m.m.f., otherwise the direct-current produces a unidirectional magnetization in the transformer, which superimposed upon the magnetic cycle raises the magnetic induction beyond satura- tion, and thus causes excessive exciting current and heating, except when very small. SYNCHRONOUS CONVERTERS 275 k *T. \ FIG. 149. Neutral of Y-connected transformers connected to neutral of three-wire system supplied from a three-phase converter. FIG. 150. Quarter-phase converter with transformer neutral connected to direct-current neutral. FIG. 151. Three-phase converter with neutral of the T-connected trans- formers as direct-current neutral. FIG. 152. Three-phase converter with transformer neutral connected to direct-current neutral. 276 ELEMENTS OF ELECTRICAL ENGINEERING For instance, with Y connection of the transformers supplying a three-phase converter, Fig. 149, each transformer secondary receives one-third of the neutral current, and if this current is not very small and comparable with the exciting current of the transformer which can rarely be the magnetic density in the transformer rises beyond saturation by this unidirectional m.m.f. This connection thus is in general not permissible for deriving the neutral. In a quarter-phase converter, as shown in Fig. 150, the trans- former neutral can be used as direct-current neutral, since in each transformer the direct current divides into two equal branches, which magnetize in opposite direction, and so neu- tralize. The T connection, Fig. 151, can be used for three-phase con- verters with the neutral derived from a point at one-third the height of the teaser transformer, as seen in Fig. 151. Delta connection on three-phase and double delta on six-phase converters cannot be used, as it has no neutral, but in this case a separate compensator is required. The diagrammatical connections of transformers can, however, be used on six-phase converters, and the connection shown in Fig. 152, which has two coils on each transformer, connected to different phases, on three-phase converters. D. ALTERNATING-CURRENT TRANSFORMER I. General 110. The alternating-current transformer consists of a magnetic circuit interlinked with two electric circuits, the primary, which receives power, and the secondary, which gives out power. Since the same magnetic flux interlinks primary and second- ary turns, the same voltage is induced in every turn of the electric circuits, and the e.m.fs. induced in the primary and in the secondary winding therefore have the ratio of turns: 'i ni r = a. . e' 2 n 2 This ratio is called the ratio of transformation. The ratio of transformation of a transformer is the ratio of turns of primary and secondary windings. In addition to the induced e.m.fs. e'i and e\, resistance r and reactance x consume voltage in primary and secondary wind- ings. The voltage consumed by the resistance represents waste of power; the voltage consumed by reactance is wattless, but causes lag of current, that is, lowers the power factor; while the in- duced voltages give the power transfer from primary to sec- ondary. Efficiency therefore requires to make the former vol- tages as small as possible, and the induced voltages as near to the terminal voltages as possible. Therefore, in first approxi- mation, the ratio of the terminal voltages e\ and e% is the ratio of transformation: As, approximately, the power output of the secondary equals the power input into the primary, it is: hence, ti 1 277 278 ELEMENTS OF ELECTRICAL ENGINEERING that is, the transformer changes from voltage e\ and current i\ to voltage 62 = and current iz = aii. In general either of the two transformer circuits may be used as primary or as secondary, and by their use transformers thus are distinguished as step-down transformers, if the primary voltage is higher than the secondary, and step-up transformers, if the secondary voltage is higher. Instead of the expression "primary" and "secondary," constructively it therefore is preferable to speak of "high voltage winding" and "low voltage winding." 111. The foremost use of the transformer therefore is for changing of the voltage: From the medium high primary distribution voltage (2300) to the low secondary consumer voltage (110, 220). From the high transmission (30 to 150 kilovolts) to the primary distribution voltage (2300) or the voltage required by syn- chronous motor, synchronous converter, etc. From the low or medium high generator voltage to the high transmission voltage. Other occasional uses of transformers are: To electrically tie systems together, so as to permit exchange of power between them, and synchronous operation. In this case, depending on the distribution of the load in the system, either transformer winding may be primary or secondary. To break up electrically a very large system, so that a ground in one part does not ground the entire system. In this case, the transformer ratio usually is 1 -f- 1. In all these cases, the transformers are "constant potential transformers," that is, primary and thus secondary voltage are constant or approximately so. Transformers supplied with constant current in the primary give practically constant current in the secondary, at a primary voltage varying with the secondary voltage. Such transformers are used in constant-current circuits, for supplying meters in high voltage circuits, etc. Further uses of transformers are for operating instruments, switches, etc., in high voltage systems. In this case, the trans- formers may be potential transformers connected across the constant voltage circuit, or current transformers connected in series into the circuit, for the supply of meters, the opera- tion of overload circuit breakers, etc. ALTERNATING-CURRENT TRANSFORMER 279 Where not expressly stated otherwise, in general a constant potential transformer is understood. II. Excitation 112. The primary current i\ is not strictly proportional to the secondary current, i 2 by the ratio of transformation, TRANSFORMER Excitation and Iron Losses Vo tage fower factor 50 30 FIG. 153. Excitation and core loss of transformer. and does not become zero at no load or open circuit, but a small and lagging current ^o remains at no load, which is called the exciting current. It produces the magnetic flux and supplies the losses in the iron, so-called "core loss." Its reactive com- ponent, i mj is called the magnetizing current, and is usually greatly distorted in wave shape, while the energy component, 280 ELEMENTS OF ELECTRICAL ENGINEERING i h , does not much differ from a sine wave, and is the hysteresis energy current: /o = ih - Jim- Under load, the primary current then consists of two com- ponents: the load current 7' 2 which is the transformed second- ary current 7' 2 = > and the exciting, current IQ. The total i primary current thus is: Ji = /' 2 + /o = ^+ (ih-jim). In general, / rarely exceeds 5 per cent, of the full-load primary current. Core loss and exciting current, with its two components, are determined by measuring volts, amperes and watts input into the primary of the transformer at open secondary. It is ob- vious that either of the transformer coils can for this purpose be used as primary, and usually the low voltage coil is employed as more convenient. Such excitation and core-loss curves are given in Fig. 153, with the impressed volts as abscissae, and the total exciting current, and core loss as ordinates. The exciting current is usually not proportional to the voltage, due to the use of a closed magnetic circuit, and for the same reason, the power-factor of the exciting current is fairly high, from 40 to 60 per cent., except at high voltages, where magnetic saturation causes an abnormal increase of the magnetizing current. The power-factor is shown on Fig. 153. IE. Losses and Efficiency 113. The losses in the transformer are (a) The core loss, comprising the loss by hysteresis and eddy currents in the iron. This depends on the maximum magnetic flux, and thus on the induced voltage: and as the induced voltage is practically equal to the impressed voltage 61, at constant impressed voltage, the core loss is practi- cally constant, and is often assumed as constant, that is, the ALTERNATING-CURRENT TRANSFORMER 281 core loss is a constant or no-load loss, and is supplied by the exciting current i . (b) The i 2 r losses in the primary and secondary coils. These are load losses, increasing with the square of the load. (c) Spurious load losses, as eddy currents in the conductors and other metal parts. With proper design these should be negligible. (d) In very high voltage transformers, electrostatic losses in the insulation appear. These usually are small in large well- designed transformers. In large transformers, the total &r loss may be less than 1 per cent., and so also the core loss, resulting in efficiencies of over 98 per cent. As instance are shown, in Figs. 154 and 155, the loss curves and the efficiency curves of two transformers, of the respective constants, at full load of 20 kw. I. Low core-loss type, Fig. 154 II. Low t*r loss type, Fig. 155 Exciting current 4 per cent. 4 per cent. Primary resistance loss 1 per cent. . 5 per cent. Secondary resistance loss Core loss 1 per cent. 1 per cent. . 5 per cent. 2 per cent. For convenience, exciting current and losses are frequently given in per cent, of the full-load output of the transformer. The curves correspond to non-inductive load. The core loss comprises hysteresis, which varies with the 1.6 power of the induced voltage and eddies proportional to the square of induced voltage. Hence, within the narrow range of variation of the induced voltage between no load and full load of a constant poten- tial transformer, the core loss can be approximated as propor- tional to the 1.7 power of the induced voltage. The induced voltage at non-inductive load equals impressed voltage minus primary ir, when neglecting the inductive drop, which is permis- sible at non-inductive load. As the induced voltage thus de- creases proportional to primary ir, the core loss decreases pro- portional to 1.7 times the primary ir. Thus, with the primary tV equal to 1 per cent, at full load, the induced voltage has 282 ELEMENTS OF ELECTRICAL ENGINEERING decreased 1 per cent, and the core loss 1.7 per cent, at full load, and correspondingly at other loads. As seen, I and II have the same full-load efficiency, but II is more efficient at overload, I at partial load. EFFICIENCY and LOSSES of Low Corcloss Transformer 1% Iron Loss 2% i 2 r Loss .9 1.0 1.1 1.2 1.3 1.4 1.5 .1 .2 .3 .4 .5 .6 .7 .8 FIG. 154. Efficiency and losses of low core loss transformer. 114. In transformers for lighting and general distribution {usually with 2300 volt primary and 2 X 115 volt secondary) the transformer is generally heavily loaded only for a short time during the day, partly loaded for a moderate time, and prac- tically unloaded for most of the time. Thus load curves of such a transformer would be: ALTERNATING-CURRENT TRANSFORMER 283 A. Lighting and power B. Lighting only 2 hours at IK load. 2 hours at IK load. 2 hours at % load. 2 hours at % load. 6 hours at Y 2 load. 20 hours of ]/ clL /2 G load. 14 hours at Ho load. EFFICIENCY and LOSSES of High Coreloss Transformer; 2 % Iron Loss 1% t 2 rLoss I V V A \ \ % \ % 100 > Loss 5.0 90 \ ,^~- \ *j 4.5 80/ /* \ s. / / 4.0 / 5 X. ^X x^ 3.5 ^> ^ ^^ ^ x^ 3.0 -^o^ *\* se^ ^^ 2^, = =r _ - ^. ^ Ire nLos ses S / ^ x 2.0 1.5 ^^ <; 1.0 j2 V-O^ ^ .5 u_i Lc ad^- ->- .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 FIG. 155. Efficiency and losses of low i l r loss transformer. 284 ELEMENTS OF ELECTRICAL ENGINEERING This gives for the two types of transformers: A. LIGHTING AND POWER Time Load = Per cent. TimeX load I II Losses Time X losses Losses Time X losses 2hr. IH 125 250 4.10 8.20 3.54 7.08 2hr. % 75 150 2.11 4.22 2.55 5.10 6hr. H 50 300 1.50 9.00 2.25 13.50 14 hr. Y20 5 S = 70 1.00 14.00 2.00 28.00 770 35.42 53.68 Input 805 . 42 823 . 68 Per cent, loss 4.41 6.51 Per cent, efficiency 95 .59 93 . 49 B. LIGHTING ONLY I II Time Load = Percent. Time X load Losses Time X losses Losses Time X losses 2hr. IK 125 250 4.10 8.20 3.54 7.08 2hr. H 75 150 2.11 4.22 2.55 5.10 20 hr. Mo 5 100 1.00 20.00 2.00 40.00 S = 500 32.42 52.18 Input 532.42 552.18 Per cent, loss 6.11 9 . 45 Per cent, efficiency 93 .89 90 . 55 As seen, while I and II have the same full-load efficiency, 97.1 per cent., I, the low core-loss type, gives a much higher all-day efficiency, the more so the shorter the time of heavy load, that is, is far preferable for general distribution, as "lighting transformer." Inversely, in large power transformers in transmission systems, the high partial load efficiency of the low core-loss type is of less importance, as such transformers are usually not run at partial load, but with a decrease of load on the system, transformers and generators are cut out and the remaining ones kept loaded. Of ALTERNATING-CURRENT TRANSFORMER 285 importance, however, is low i 2 r loss. Under emergency conditions requiring overloading of some transformer, the increased loss is all in the copper, and the less therefore the i 2 r y the less is the danger of destruction by overheating in a case of a temporary overload. Thus the low i z r loss type of transformer is preferable for large power units. IV. Regulation 115. As primary and secondary winding of the transformer can- not occupy the same space, and in addition some insulation more or less depending on the voltage must be between them, there is thus a space between primary and secondary through which the primary current can send magnetic flux which does not interlink with the secondary winding, but is a self-induc- tive or leakage flux and in the same manner the secondary current sends self-inductive or leakage flux through the space between primary and secondary winding. These fluxes give rise to the self -inductive or leakage reactances x\ and Xz of the transformer. Or in other words, two paths exist for magnetic flux in the transformer: the path surrounding primary and secondary coils, through which flows the mutual magnetic flux of the transformer, which is the useful flux, that is, the flux which transfers the power from primary to secondary circuit; and the space between pri- mary and secondary winding through which the self-inductive or leakage flux passes, that is, the flux interlinked with one wind- ing only, but not the other one. The latter flux thus does not transmit power, but consumes reactive voltage and thereby pro- duces a voltage drop and a lag of the current behind the voltage, that is, is in general objectionable. The mutual magnetic flux passes through a closed magnetic circuit, with the (vector) difference between primary and second- ary current, that is, the exciting current J = /i as m.m.f. The self-inductive flux passes through an open magnetic circuit of high reluctance, the narrow space between primary and secondary windings, but it is due to the full m.m.f. of primary or secondary current and, therefore, in spite of the high reluctance of the leakage flux path due to the high m.m.f. (20 times as great as that of the mutual flux at 5 per cent, exciting current), this flux and the reactance voltages caused by it are appreci- 286 ELEMENTS OF ELECTRICAL ENGINEERING able, usually between 2 per cent, and 8 per cent, in modern transformers. The distribution of the leakage flux between primary and secondary winding, that is, between primary reactance x\ and secondary Xz, is to some extent arbitrary (see discussion in "Theory and Calculation of Electric Circuits'')) and the methods of test give only the sum of the primary and the secondary re- actance, the latter reduced to the primary by the ratio of trans- formation : Xi + a 2 x 2 . 116. The total reactance of primary and secondary, and also TRANSFORMER I mpedance and Short Circuit Losses 7 .1 .2 .3 .1 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 l.i 1.5 FIG. 156. Impedance and short circuit losses of transformer. the total (effective) resistance of primary and secondary winding are measured by impressing voltage on the primary coil, with the secondary winding short-circuited, and measuring volts, amperes and watts. In this test the voltage usually is impressed upon the high voltage winding, as the impedance voltage is only a small part of the operating voltage of the transformer. Such "impedance curves" and "short-circuit loss curves" for the transformers in Figs. 154 and 155 are shown in Fig. 156. If the short-circuit loss is greater than the sum of primary and ALTERNA TING-CURRENT TRANSFORMER 287 secondary i z r losses, the difference represents load losses caused by eddy currents in the conductors, etc. * The reactance of the transformer is often given as percentage. Six per cent, reactance thus means that the primary ix, as per cent, of the primary impressed voltage, plus the secondary ix as per cent, of the secondary voltage, is 6 per cent. Or: *'*' + **'*'= 0.06. e\ Especially since x\ and x z cannot be separated experimentally, but the impedance test gives the sum of primary reactance x\, and secondary reactance x 2 reduced to the primary by the ratio of transformation a, that is this is permissible. The foremost effects of the leakage reactance of the trans- former are, to affect the voltage regulation, and to determine the short-circuit current and the mechanical forces resulting from it. 117. The exciting current, being a small and practically con- stant component of the primary current, does not affect the regu- lation of the transformer appreciably, and thus can be neglected in the calculation of the regulation curve. If this is done, the secondary quantities can be reduced to the primary by the ratio of transformation (or inversely), that is, by multiplying all secondary voltages and dividing all secondary currents by a, and multiplying all secondary impedances by a 2 , or inversely when reducing from primary to secondary. 1 Or, primary and secondary impedances can be given in per cent., that is, the primary ir and ix in per cent, of the primary voltage, the secondary ir and ix in per cent, of the secondary voltage, and in this case, primary and secondary quantities can be directly added. This usually is the most convenient way, at least for approximate calculation. Thus in the transformer shown in Fig. 154, let = 0.02 be the total reactance (2 per cent.), at full non- inductive load. 1 As the transformation ratio of the voltage is a, that of the current is -i the transformation ratio of the impedances (resistance and reactance), is volts a 2 , as impedance = amperes 288 ELEMENTS OF ELECTRICAL ENGINEERING p = 0.02 is the total resistance, primary and secondary com- bined. At the percentage p of the non-inductive load, the voltage consumed by reactance is p% = 0.02 p and in quadrature with the current and thus with the voltage at non-inductive load, hence subtracts by ^/difference of squares: while the voltage consumed by the resistance is pp 0.02 p and in phase with the voltage, hence directly subtracts, leaving: - P 2 ? ~ PP = V 1 - 0.0004 p 2 - 0.02 p as the voltage at percentage p of load, given as per cent, of the open-circuit or no-load voltage. The voltage drop at-frac- REGULATION of TRANSFORMER Non-inductive Load; I Z = .02 + .02j II .01 + .04 j III .01 + .08 j 3.5 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 FIG. 157. Regulation curve of transformer: non-inductive load. tional load p, as fraction of full-load voltage, that is, the regula- tion of the transformer at non-inductiVe load, then is R = 1 - VI - p 2 ^ + PP = 1 - V 1 - 0.0004 p 2 + 0.02 p or, resolved by the binomial, and dropping the higher terms: R = PP + \ P 2 ? = 0.02 p + 0.0002 p 2 = P (P + \ P?} = 0.02 p (1 + 0.01 p) As curves I, II, III in Fig. 157 are shown the regulation curves of three transformers: ALTERNATING-CURRENT TRANSFORMER 289 I: 2 per cent, resistance and 2 per cent, reactance. II : 1 per cent, resistance and 4 per cent, reactance. Ill : 1 per cent, resistance and 8 per cent, reactance. FIG. 158. Vector diagram of transformer regulation. % 6.5, REGULATION of TRANSFORMER Inductive Load: 20 Lag I Z = .02 + .02 3 II .01 + .04 3 III .01 + .08 3 / <<* / / 5^ / / 5.0 / / 4^ / S 4.^ 1 7 ^^f ^ / x^ s^ 3.0 / / Ix ^ s^ 2.5 / / <> ^1 2.0 / ^ t> ^ 1^ S / /? ^ 1.0 / ^ ^ X X Lc iad : - .1 ^ .3 .4 ^ .6 .7 .8 .9 1.0 LI LI U 1.4 U FIG. 159. Regulation of transformer, moderately inductive load. Calculated respectively by the equations given at end of next paragraph. 118. At inductive load of power-factor cos o>, that is, the lag of the current behind the voltage by angle w, the regulation 290 ELEMENTS OF ELECTRICAL ENGINEERING curve is derived from the vector diagram Fig. 158. The ir voltage is in phase with the current, the ix voltage 90 deg. ahead of the current. Resolving both of these voltages into components in phase and in quadrature with the terminal voltage, gives (Fig. 158): 16.0 / % / 10.0 REGULATION of TRANSFORMER Inductive Load: 60 Lag I Z = .02 -f .02 j II .01+.04J III .01+ .08 j / / 9.5 / 9.0 / 8.5 / / 8.0 / 7.5 / 7.0 / / 6.5 III / ^ / / 5.5 / / / / 5.0 / / / 4.5 / ^ _x ^^O / / / X x^ 3.5 / / / I X x 3.0 / / / x X 2.5 / / /I x / 2.0 / / .X X 1.5 / / / ^ 1.0 / 'x S x^ Loa d:_ >- .5 ^ X .1 .2 .3 .4 .6 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 FIG. 160. Regulation of transformer, highly inductive load. ir cos co and ix sin co in phase with e, ix cos co and ir sin co in quadrature with e. The former thus directly subtract, and the latter subtract by A/difference of squares, thus giving as resultant voltage : (ix cos co ir sin co) 2 (ir cos co + ix sin co) ALTERNATING-CURRENT TRANSFORMER 291 or, since ir at full load as fraction of e is p, and ix as fraction of 6 is ; at the fraction p of the load: ir = pp, ix = p, the re- sultant voltage is: \/l p 2 (^ cos co p sin co) 2 P (p cos co + sin co) and the regulation of the transformer, at inductive load of angle of lag co, thus is- R = 1 A/1 p 2 ( cos co p sin co) 2 + p (p cos co + sin co). Resolving again the square root by the binomial, and arrang- ing, gives, by dropping out terms of higher order: v 2 R p (p cos co + sin co) + ~ ( cos co p sin co) 2 In Figs. 159 and 160 are shown, for the angles of lag co = 20 (moderately inductive load, 94 per cent, power-factor), and co = 60 (highly inductive load, 50 per cent, power-factor), the regulation of the same three transformers as in Fig. 157, cal- culated respectively from the expression: REGULATION OF TRANSFORMERS Per cent, resistance, p = 0.02 0.01 0.01 Per cent, reactance, = 0.02 0.04 0.08 ^ w of lag: Curve I Curve II Curve III Fig. 157, R = 0.02p O.Olp O.Olp (1 +0.01p) (1 +0.08p) (1 +0.32p) Fig. 159, 20 R = 0.0256p 0.0231p 0.0368p (1 + 0.0027p) (1 + 0.025p) (1 +0.07?) Fig. 160, 60 R = 0.0273p 0.0396p 0.0743p (! 4 +0.001p) (1 + 0.0016p) (1+0. 0066 p) Non-inductive 20 lag. 60 lag. P I II III I II III I II III 0.2 0.40 0.20 0.21 0.51 0.46 0.75 0.55 0.79 1.49 0.4 0.80 0.41 0.45 1.02 0.93 1.52 1.09 1.58 2.99 0.6 1.20 0.63 0.71 1.54 1.40 2.30 1.64 2.38 4.48 0.8 1.61 0.85 1.00 2.05 1.88 3.13 2.18 3.17 5.98 1.0 2.02 1.08 1.32 2.56 2.37 3.94 2.73 3.96 7.48 1.2 2.43 1.31 1.66 3.08 2.85 4.79 3.28 4.76 9.00 1.4 2.84 1.56 2.03 3.59 3.34 5.67 3.83 5.56 10.50 1.6 3.25 1.80 2.42 4.10 3.83 6.56 4.38 6.35 12.00 i 292 ELEMENTS OF ELECTRICAL ENGINEERING 119. As seen, at non-inductive load, Fig. 157, the reactance of the transformer, even if fairly high, has practically no effect, but the resistance controls the regulation. At moderately inductive load reactance as well as resistance affect the regulation; doubling the reactance while halving the resistance, gives practically the same regulation. FIG. 161. High reactance transformer construction. At highly inductive load the reactance of the transformer be- gins to predominate over the resistance in affecting the regulation. Thus, where close regulation is required, as in lighting and general distribution transformers, low reactance is of impor- tance. This is given by reducing the section of the leakage path that is, bringing primary and secondary windings as close together FIG. 162. Low reactance transformer construction. as possible and by reducing the m.m.f. which produces the leak- age flux, by subdividing primary and secondary winding into a number of coils and intermixing these coils, so that the leakage flux of each path is due to a small part of the total m.m.f. of primary or secondary only, as shown in Figs. 161 and 162. In Fig. 162 the m.m.f. of each of the four leakage paths is due to ALTERNATING-CURRENT TRANSFORMER 293 one-fourth of the m.m.f. as in Fig. 161, and the leakage flux density thus reduced to one-fourth of what it is in Fig. 161. As furthermore the section of each leakage flux in Fig. 162 is materially less than in Fig. 161, due to the lesser thickness of the coils, it follows that in Fig. 162 the leakage flux interlinked with each turn of each winding, and thus the reactance of the transformer, is materially less than one-quarter of what it is in Fig. 161. The regulation of the transformer at anti-inductive load, that is, for leading secondary current, obviously is given by the same equation as that for lagging current, by merely substituting co for co. V. Short-circuit Current 120. If a short circuit occurs at the secondary terminals of a transformer, and the power supply at the primary is sufficient to maintain the primary terminal voltage, the primary and second- ary currents of the transformer are limited by its impedance only. Thus, if r = P + j* is the impedance voltage, as fraction of full-load voltage, the short- circuit current of the transformer is 1 1 of the full-load current, thus usually is very large. In the three instances illustrated in Figs. 157, 159 and 160, with f = 0.02 + 0.02 j, hence f =0.028 0.01 + 0.04 j 0.04 0.01 + 0.08 j 0.08 the short-circuit current thus is 36, 25 and 12.5 times full-load current, respectively. As seen, with the exception of very low reactance transformers, it is essentially the reactance which determines the total im- pedance and thus the short-circuit current. 121. Primary current and secondary current in the trans- former, being opposite in phase, repel each other. This repul- sion is proportional to the product of primary and secondary current, thus, since primary and secondary current are (ap- 19 294 ELEMENTS OF ELECTRICAL ENGINEERING proximately) proportional to each other, the repulsion is pro- portional to the square of the current. The repulsion is small at full load, but in low-reactance transformers, with 'short-circuit currents from forty to fifty times full-load current, the mechanical forces have increased 1600 to 2500 fold, and then, with large power transformers, reach formidable values, amounting to many hundred tons, and then it is economically difficult to build trans- formers with the coils supported so rigidly as to stand such forces. Thus far very few generating systems exist of such large size as to be capable of maintaining full voltage at the primary ter- minals of a large transformer at secondary short circuit, but their number is increasing, and thus the necessity of limiting the short- circuit current of large power transformers to a mechanically safe value is becoming increasingly important. This means a construction providing for considerable internal reactance. As the regulation of large power transformers is of no serious impor- tance, the desirability of low reactance, which exists in the small lighting and general distribution transformers, does not exist in large power transformers, and modern practice tends toward the use of internal reactance of 4 to 8 per cent., to secure reasonable mechanical safety. VI. Heating and Ventilation 122. As the transformer is a stationary apparatus, it does not have the advantage of dissipating the heat produced by the internal losses, by the natural ventilation of the air currents pro- duced by the centrifugal forces in rotating apparatus, and it is therefore fortunate that the transformer is the most efficient apparatus (except perhaps the electrostatic condenser) and thus has to dissipate less heat than any other apparatus of the same output. Thus in smaller transformers radiation and the natural convection from the surface are often sufficient to keep the tem- perature within safe limits. Smaller distribution transformers usually are installed out- doors, on poles, and then require protection by enclosure in an iron case or tank. This still further reduces the heat radiation, and therefore such transformer cases are now almost always filled with oil, the oil serving to carry the heat from the transformer iron and windings to the case. Incidentally, the oil filling also protects the transformer from the failure of insulation by con- ALTERNATING-CURRENT TRANSFORMER 295 densation of moisture during the variation of atmospheric tem- perature and humidity. In larger oil-cooled transformers, the tank is made corrugated, even with large double corrugations, to give a very large external surface to dissipate the heat. Much more effectively, however, the heat can be carried away by mechanical ventilation, and size and cost of the trans- former thereby materially reduced. Therefore practically all larger transformers have forced ventilation. Various methods of forced ventilation are: (a) Oil circulation. The warm oil is pumped from the top of the transformer tank, through some cooling device. Often also a drying device to take out any trace of moisture and then fed back into the bottom of the tank. (6) Water circulation. Cooling water is pumped through a system of pipes located under the oil at the top of the trans- former tank. This is the most common design of large trans- formers. (c) Air blast. Coils and iron are subdivided by ventilating ducts, and a low-pressure air blast forced through the ventilating ducts. This is the cleanest method, as no oil is used. However, it is limited to low and moderate voltages up to about 33,000; at higher voltage, the mechanical and chemical action of corona appearing at the coils reduces their life, and the oil becomes necessary for insulation. Numerous modifications of the various types have been built and are in use, as water-cooled oil transformers with natural circulation of the water through outside radiating pipes, etc. VH. Types of Transformers 123. As the transformer consists of a magnetic circuit inter- linked with two electric circuits, two constructive arrangements are possible : The electric circuits may be inside, and surrounded by the magnetic circuit as shell, shell-type transformer; or the magnetic circuit may be arranged inside, as core, and sur- rounded by the electric circuits, core-type transformer. In their simplest form, Fig. 163 shows diagrammatically the core-type transformer, with the iron Fe as inside circular core, built up of laminations or of iron wire, and the windings Cu outside; Fig. 164 shows diagrammatically the shell-type 296 ELEMENTS OF ELECTRICAL ENGINEERING transformer, with the copper windings inside, as Cu, and the iron shell Fe wound around it, of iron wire, etc. However, the circular form 163 is used to a limited extent only, in small trans- formers, autotransformers and reactances, and the form 164 practically never used, and in the constructive modification from these diagrammatic types, it is often difficult to decide to which type to assign the transformer. FIG. 163. Diagram of core type transformer. FIG. 166. Diagram of shell type transformer. The typical shell-type transformer of today is shown in section in Fig. 165, with the magnetic circuit Fe, and the high voltage windings P and low- voltage windings S intermixed with each other. Core-type transformers are shown in section in Figs. 166 and 167, the former with one, the latter with two cores, and with two different coil arrangements, the intermixed and the concentric. ALTERNATING-CURRENT TRANSFORMER 297 For the transformation of three-phase circuits, three separate single-phase transformers may be used, and their primaries and FIG. 165. Shell type transformer. FIG. 166. Single-coil core type transformer. FIG. 167. Two coil core type transformer. FIG. 168. Shell type three-phase FIG. 169. Core type three-phase transformer diagram. transformer diagram. secondaries then connected in ring or delta connection or in star or Y connection, giving the four arrangements: AA, AF, FA, YY. Or two transformers may be used, arranged in T connection or in open A connection, as further discussed under three-phase systems. Or a three-phase transformer may be used. Diagram- 298 ELEMENTS OF ELECTRICAL ENGINEERING matically, the three-phase transformer can be represented by Fig. 168, shell type, and Fig. 169, core type. 124. While in its magnetic and electrical characteristics there is no essential difference between the single-phase shell- type and the single-phase core-type transformer, there is a material difference in the three-phase transformer. In the shell type, Fig. 168, a short circuit of one of the three phases does not affect the magnetic and thus the electric circuit of the other two phases, in the core type Fig. 169, however, a short circuit of one of the three phases short circuits the magnetic return of the other two phases, and so acts as a partial electrical short circuit of these two other phases. In shell-type transformers, Fig. 168, a triple harmonic of flux can exist, but not in the core type, Fig. 169. In the three- FIG. 170. Shell type three-phase transformer. phase system, the three voltages, currents, etc., are displaced in phase from each other by 120. Their third harmonics therefore are displaced in phase from each other by 3 X 120, that is, by 360, or in other words, are in phase with each other. In Fig. 169, such triple frequency fluxes in the three cores would have no magnetic return, except by leakage through the air, that is, cannot exist, except in negligible intensity, and there- fore the core type of three-phase transformer cannot give any serious triple frequency voltage. In the shell type Fig. 168, however, the three triple frequency fluxes, being in phase with each other, produce a triple frequency single-phase flux through a closed magnetic circuit. Where the circuit conditions and connections are such as to give a triple harmonic as with YY connection the shell-type three-phase transformer may produce triple frequency voltages, resulting from the triple frequency ALTERNATING-CURRENT TRANSFORMER 299 flux, and under unfavorable conditions, as when connecting to a system of high capacity which intensifies these voltages this may lead to destructive voltages, and YY connections with shell-type three-phase transformers thus lead to serious high voltage dangers. 125. The usual shell-type construction of three-phase trans- formers is shown in section in Fig. 170, the core type in Fig. 171. In Fig. 170 economy requires that the middle phase is con- nected in opposite direction to the outside phases, so that the iron between the successive phases, at 1, 2 and 2, 3, carries the sum of two of the three-phase fluxes, which, as the fluxes are 120 deg. apart, equals one of the fluxes. If the middle phase were not reversed, 1, 2 and 2, 3 would carry the difference of II II FIG. 171. Core type three-phase transformer. two fluxes 120 deg. apart, and this difference is V3 times each flux, thus would give a much higher loss. In Fig. 171 usually the exciting current of the middle phase is somewhat less than that of the outside phase, since the magnetic reluctance of the middle phase is slightly lower. VIII. Autotransformer 126. If in a transformer a part of the secondary winding is used as primary, or inversely, the transformer is an autotrans- former, sometimes also called compensator. Thus let in a transformer Fig. 172 primary current, voltage and turns be respectively ii, e^ ni, and secondary current, voltage and turns be t' 2 , e 2 , n 2 , thus the ratio of transformation a = n 2 Assuming HI > n z , then in any n 2 of the HI primary turns, the same voltage is induced as in the n 2 secondary turns, and we could thus 300 ELEMENTS OF ELECTRICAL ENGINEERING use any n 2 primary turns as secondary turns, provided we make them of sufficient copper section to carry the secondary current. The n 2 turns in Fig. 173 thus are in common to primary and secondary circuit. As primary and secondary current are (ap- proximately) opposite in phase, the current in the common turns of Fig. 173 is (approximately, that is, neglecting exciting current) the difference between secondary and primary current, i 2 ii, thus less than the secondary current i' 2 , and as the result, the com- mon turns in Fig. 173 may be made of less copper section than the secondary turns in Fig. 172, while the number of primary turns is reduced by n 2 . Thus an autotransformer requires less copper, that is, is smaller and cheaper than a transformer of the same output. 127. In the transformer Fig. 172, the size is determined by the number of turns and turn sections, that is, by e\ X ii + 2 X iz FIG. 172. Diagram of trans- former. FIG. 173. Diagram of auto- transformer. (the turns being proportional to the voltage, the turn section to the current, the same magnetic flux assumed). But since 61 = ae z and i\ = , e\i\ = e 2 i 2 , and the size of the transformer Fig. 172 thus is proportional to 2 e-# 2 , that is, to 2 P, or twice the output. In the autotransformer Fig. 173, the n z common turns are tra- versed by the difference of secondary and primary current, at secondary voltage, and the size of this common part of the wind- ing thus is: 62 (iz ii). The remaining part of the winding, of n\ n 2 turns, that is, of voltage e\ e 2) is traversed by the primary current ii, hence of size i\ (e\ e 2 ), and the total size of the autotransformer thus is : 62 (*2 ii) + i\ (e\ e 2 ) ALTERNATING-CURRENT TRANSFORMER 301 but, substituting again for ii and ei, gives as the size of the auto- transformer: (ae2 - es) = 2 -"('-3 hence, the ratio of size of autotransformer and of transformer of the same output, is: _ autotransformer _ 1. transformer a If the ratio a = 2, as transforming between 115 and 230 volts, 7 = J, that is, the autotransformer has half the size of the trans- former, or, more correctly stated, the autotransformer is as large as a transformer of half the output. If the ratio a = 1.1, as raising (or lowering) the voltage 10 per cent, by autotransformer, this autotransformer has the size 7 = 0.1 that is, is as small as a transformer of one-tenth the out- put. If the ratio a = 10, as transforming between 2300 and 230, 7 = 0.9, that is, the autotransformer is only 10 per cent, smaller than the transformer. The saving in size and therewith in efficiency and cost by the use of the autotransformer thus is the greater, the lower the transformation ratio a, but becomes negligible at high trans- formation ratios. Thus autotransf ormers are very economical for use in moderate voltage transformation, as a voltage change by 10 or 20 per cent., or even for doubling the voltage, or dividing it in two, but not for high voltage ratios. 128. The most serious disadvantage of the autotransformer obviously is that it electrically interconnects primary and sec- ondary circuit and thereby puts the voltage of the higher voltage circuit onto the lower voltage circuit. Thus, when using auto- transformers, the insulation of the low voltage circuit and the high potential tests of all the apparatus used in the low voltage circuit must be those of the high voltage circuit. Furthermore, a ground in one of the two circuits of an autotransformer also is a ground on the other circuit, while with a transformer, a ground on the secondary does not ground the primary, and in- versely. With low voltages, as 115 -5- 230 volt transformation, this is usually of no importance. It would be a serious objection 302 ELEMENTS OF ELECTRICAL ENGINEERING when attempting the use of autotransformers between 2300 and 230 volts. For instance, a ground at the off side of the high- voltage winding, at A in Fig. 173, would put the entire secondary winding 2300 to 2070 volts above ground, and thus the secondary circuit would kill anybody who touches it while standing on the ground. Any transformer of voltage e\ and 62 and currents i\,i% can be converted into an autotransformer, by connecting primary and secondary in series, of voltages e\ + e z and e z and currents i\ and i z + i\. And inversely, any autotransformer, by disconnect- ing the two sections of the coil, would give (provided that the insulation is sufficient) a transformer of (ei e^) X i\ primary, and e z X (i* ii) secondary circuit. The regulation of an autotransformer is better, and the effi- ciency higher, than that of the same structure as transformer, and the per cent, reactance lower, that is the short-circuit current higher in the autotransformer than in the same structure as transformer. Very often it is difficult to build autotransformers with sufficiently high internal reactance, to make them safe under momentary short circuit as autotransformers, while they may be . perfectly safe as transformers, where the reactance is higher. This is a serious objection to the use of autotransformers in high-power systems. IX. Reactors (Reactive Coils, Reactances) 129. The reactor consists of one electric circuit interlinked with a magnetic circuit, and its purpose is, not to transform power, but to produce wattless or reactive power, that is, lagging current, or what amounts to the same, leading voltage. While therefore theoretically we cannot speak of an ''efficiency" of a reactor, since there is no power output, nevertheless in the in- dustry the expression " efficiency of a reactive coil" is gener- ally used, and generally understood, in the conventional definition : T^C 1 SS Efficiency = 1 -. input and the input is given in total volt-amperes, the loss in energy volt-amperes, that is, watts. The efficiency then is 1 power- factor. ALTERNATING-CURRENT TRANSFORMER 303 The transformer at open circuit is a reactor, but a very poor one, as its power-factor is high, that is, the efficiency low. In the transformer, the exciting ampere-turns are the (vector) difference between primary and secondary ampere-turns, are wasted, and therefore made as low as possible, by using a closed magnetic circuit. In the reactor, no secondary circuit exists, but the exciting ampere-turns are the purpose of the device, thus should be as large as possible. That is, to convert a trans- former into a reactor, the reluctance of the magnetic circuit must be increased so as to make the exciting ampere-turns equal to the total full-load ampere-turns of the structure as transformer. This is done by inserting an air gap into the magnetic circuit. Such a gap may be either a single gap, or a number of smaller air gaps, or one or a number of slots cutting almost through the magnetic circuit, but leaving narrow bridges, FIG. 174. Bridged air-gap reactor. as shown in Fig. 174. This latter offers the advantage of a better mechanical structure, less liability to noise and to magnetic leakage, but when used in series in high voltage circuits, may lead to voltage peaks at the moment of current reversal, which may endanger the insulation. The use of a number of small air gaps instead of one large one distributes the magnetic leakage and thus gives less liability to eddy currents in the conductors. 130. A transformer of output P = e 2 i z has a size of winding space of e z i2 + #iii = 2 e 2 z*2, that is (with the air gap inserted into the magnetic circuit), gives a reactor of the capacity ei = 2 P. That is, a reactor has the size of a transformer of half its output. Reactors are frequently used in series to apparatus, and the vol- tage consumed by the reactance then varies with the current, and is, due to the air gap, proportional to the current up to the value where the iron part of the reactance begins to saturate, as shown by the characteristic curve of a reactance, Fig. 175, the "volt- 304 ELEMENTS OF ELECTRICAL ENGINEERING ampere characteristic." Then the voltage increases less than proportional to the current, or inversely, the current increases out of proportion to the voltage, that is, the reactance decreases and wave-shape distortion occurs. Reactances thus must be designed so that at the highest currents (or voltages), at which they may be called upon to develop their reactance, their magnetic circuit is still below saturation. Industrially, reactors are often denoted in per cent. Thus for Volt- REACTOR Ampere Characteristic Rec Vo ts e FIG. 175. Volt-ampere characteristic of reactor. phase control in synchronous converter circuits, 15 per cent, re- actances are used. This means, at full-load current, the voltage consumed by these reactances is 15 per cent, of the circuit voltage. 131. With the increasing size and increasing voltage of modern central stations and the use of high-speed turbo-alternators ca- pable of momentarily giving very high short-circuit currents, the amount of power, which can be developed momentarily by a short circuit in the system near the generating station, has reached such ALTERNATING-CURRENT TRANSFORMER 305 destructive values, that a limitation of this power has become necessary, and as economy of operation forbids sectionalizing the system into a number of smaller units, this has led to the exten- sive use of power-limiting reactances, in the generator leads, in the bus bars, tie feeders and even the power feeders. Such re- actances are used of 2' to 8 per cent., and in bus bars even up to 25 per cent., and in case of a local short circuit, limit the current which can flow. Thus a 4 per cent, reactance would at a short circuit just beyond the reactance limit the current to -r - = 4 per cent. 25 times the normal, etc. But to do so, the reactance must still be there at twenty-five times its rated current, that is, when ab- sorbing full circuit voltage instead of its normal 4 per cent, thereof. If then iron is used in the magnetic circuit of such a reactance, the density must be so low, that at twenty-five times this density (or at 12.5 times with an 8 per cent, reactance, etc.), it does not yet saturate. When limited to such very low mag- netic densities in the iron, the mass of iron becomes so enormous, that it becomes more economical to use an air circuit throughout. Reactances, which must retain their reactance, that is, must not saturate at many times their normal current, such as power limiting reactances, thus are built without iron in the magnetic circuit. E. INDUCTION MACHINES I. General 132. The direction of rotation of a direct-current motor, whether shunt- or series-wound, is independent of the direction of the current supplied thereto; that is, when reversing the current in a direct-current motor the direction of rotation remains the same. Thus theoretically any continuous-current motor should operate also with alternating currents. Obviously in this case not only the armature but also the magnetic field of the motor must be thoroughly laminated to exclude eddy currents, and care taken that the currents in the field and armature circuits reverse simultaneously. Obviously the simplest way of fulfilling the latter condition is to connect the field and armature circuits in series as alternating-current series motor. Such motors are used to a considerable extent, but, like the shunt motor, have the dis- advantage of a commutator carrying alternating currents. The shunt motor on an alternating-current circuit has the objection that in the armature winding the current should be power current, thus in phas with the e.m.f., while in the field winding the current is lagging nearly 90 deg., as magnetizing current. Thus field and armature would be out of phase with each other. To overcome this objection either there is inserted in series with the field circuit a condenser of such capacity as to bring the current back into p>hase with the voltage, or the field may be excited from a separate e.m.f. differing 90 deg. in phase from that supplied to the armature. The former arrange- ment has the disadvantage of requiring almost perfect con- stancy of frequency, and therefore is not practicable. In the latter arrangement the armature winding of the motor is fed by one, the field winding by the other phase of a quarter-phase sys- tem, and thus the current in the armature brought approximately into phase with the magnetic flux of the field. Such an arrangement obviously loads the two phases of the system unsymmetrically, the one with the armature power current, the other with the lagging field current. To balance the system two such motors may be used simultaneously and 306 INDUCTION MACHINES 307 combined in one structure, the one receiving power current from the first, magnetizing current from the second phase, the second motor receiving magnetizing current from the first and power current from the second phase. The objection that the use of the commutator is complicated and greatly limits the design to avoid serious sparking can be en- tirely overcome by utilizing the alternating feature of the current ; that is, instead of leading the current into the armature by com- mutator and brushes, producing it therein by electromagnetic induction, by closing the armature conductors upon themselves and surrounding the armature by a primary coil at right angles to the field exciting coil. Such motors have been built, consisting of two structures each containing a magnetizing circuit acted upon by one phase and a primary power circuit acting upon a closed-circuit armature as secondary and excited by the other, phase of a quarter-phase system (Stanley motor) . Going still a step further, the two structures can be com- bined into one by having each of the two coils fulfill the double function of magnetizing the field and producing currents in the secondary which are acted upon by the magnetization produced by the other phase. Obviously, instead of two phases in quadrature any number of phases can be used. This leads us by gradual steps of development from the con- tinuous-current shunt motor to the alternating-current polyphase induction motor. In its general behavior the alternating-current induction motor is therefore analogous to the continuous-current shunt motor. Like the shunt motor, it operates at approximately constant mag- netic density. It runs at fairly constant speed, slowing down gradually with increasing load. The main difference is that in the induction motor the current in the armature does not pass through a system of brushes, as in the continuous-current shunt motor, but is produced in the armature as the short-circuited secondary of a transformer; and in consequence thereof the primary circuit of the induction motor fulfills the double func- tion of an exciting circuit corresponding to the field circuit of the continuous-current machine and a primary circuit produc- ing a secondary current in the secondary by electromagnetic induction. 308 ELEMENTS OF ELECTRICAL ENGINEERING 133. Since in the secondary of the induction motor the cur- rents are producjed by induction from the primary impressed currents, the induction motor in its electromagnetic features is essentially a transformer; that is, it consists of a magnetic cir- cuit or magnetic circuits interlinked with two electric circuits or sets of circuits, the primary and the secondary circuits. The difference between transformer and induction motor is that in the former the secondary is fixed regarding the primary, and the electric energy in the secondary is made use of, while in the latter the secondary is movable regarding the primary, and the me- chanical force acting between primary and secondary is used. In consequence thereof the frequency of the currents in the sec- ondary of the induction motor differs from, and as a rule is very much lower than, that of the currents impressed upon the pri- mary, and thus the ratio of e.m.fs. generated in primary and in secondary is not the ratio of their respective turns, but is the ratio of the product of turns and frequency. Taking due consideration of this difference of frequency be- tween primary and secondary, the theoretical investigation of the induction motor corresponds to that of the stationary trans- former. The transformer feature of the induction motor pre- dominates to such an extent that in theoretical investigation the induction motor is best treated as a transformer, and the elec- trical output of the transformer corresponds to the mechanical output of the induction motor. The secondary of the motor consists of two or more circuits displaced in phase from each other so as to offer a closed sec- ondary to the primary circuits, irrespective of the relative motion. The primary consists of one or several circuits. In consequence of the relative motion of the primary and secondary, the magnetic circuit of the induction motor must be arranged so that the secondary while revolving does not leave the magnetic field of force. That means, the magnetic field of force must be of constant intensity in all directions, or, in other words, the component of magnetic flux in any direction in space be of the same or approximately the same intensity but differing in phase. Such a magnetic field can either be considered as the superposition of two magnetic fields of equal intensity in quad- rature in time and space, or it can be represented theoretically by a revolving magnetic flux of constant intensity, or rotating INDUCTION MACHINES 309 field, or simply treated as alternating magnetic flux of the same intensity in every direction. 134. The operation of the induction motor thus can also be considered as due to the action of a rotating magnetic field upon a system of short-circuited conductors. In the motor field or primary, usually the stator, by a system of polyphase impressed e.m.fs. or by the combination of a single-phase impressed e.m.f. and the reaction of the currents produced in the secondary, a rotating magnetic field is produced. This rotating field produces currents in the short-circuited armature or secondary winding, usually the rotor, and by its action on these currents drags along the secondary conductors, and thus speeds up the armature and tends to bring it up to synchronism, that is, to the same speed as the rotating field, at which speed the secondary currents would disappear by the armature conductors moving together with the rotating field, and thus cutting no lines of force. The secondary therefore slips in speed behind the speed of the rotating field by as much as is required to produce the secondary currents and give the torque necessary to carry the load. The slip of the induction motor thus increases with increase of load, and is approximately proportional thereto. Inversely, if the secondary is driven at a higher speed than that of the rotating field, the field drags the armature conductors back, that is, consumes mechanical torque, and the machine then acts as a brake or induction generator. In the polyphase induction motor this magnetic field is pro- duced by a number of electric circuits relatively displaced in space, and excited by currents having the same displacement in phase as the exciting coils have in space. In the single-phase motor one of the two superimposed mag- netic quadrature fields is excited by the primary electric circuit, the other by the . secondary currents carried into quadrature position by the rotation of the secondary. In either case, at or near synchronism the magnetic fields are practically identical. The transformer feature being predominant, in theoretical investigations of induction motors it is generally preferable to start therefrom. The characteristics of the transformer are independent of the ratio of transformation, other things being equal; that is, dou- bling the number of turns for instance, and at the same time reducing their cross section to one-half, leaves the efficiency, regulation, etc., of the transformer unchanged. In the same way, 310 ELEMENTS OF ELECTRICAL ENGINEERING in the induction motor it is unessential what the ratio of primary to secondary turns is, or, in other words, the secondary circuit can be wound for any suitable number of turns, provided the same total copper cross section is used. In consequence hereof the secondary circuit is mostly wound with one or two bars per slot, to get maximum amount of copper, that is, minimum resist- ance of secondary. The general characteristics of the induction motor being inde- pendent of the ratio of turns, it is for theoretical considera- tions simpler to assume the secondary motor circuits reduced to the same number of turns and phases as the primary, or of the ratio of transformation 1 to 1, by multiplying all secondary cur- rents and dividing all secondary e.m.fs. by the ratio of turns, multiplying all secondary impedances and dividing all secondary admittances by the square of the ratio of turns, etc. Thus in the following under secondary current, e.m.f., impe- dance, etc., shall always be understood their values reduced to the primary, or corresponding to a ratio of turns 1 to 1, and the same number of secondary as primary phases, although in prac- tice a ratio 1 to 1 will hardly ever be used, as not fulfilling the condition of uniform effective reluctance desirable in the start- ing of the induction motor. II. Polyphase Induction Motor 1. INTRODUCTION 135. The typical induction motor is the polyphase motor. By gradual development from the direct-current shunt motor we arrive at the polyphase induction motor. The magnetic field of any induction motor, whether supplied by polyphase, monocyclic, or single-phase e.m.f., is at normal condition of operation, that is, near synchronism, a polyphase field. Thus to a certain extent all induction motors can be called polyphase machines. When supplied with a polyphase system of e.m.fs. the internal reactions of the induction motor are simplest and only those of a transformer with moving second- ary, while in the single-phase induction motor at the same time a phase transformation occurs, the second or magnetizing phase being produced from the impressed phase of e.m.f. by the rota- tion of the motor, which carries the secondary currents into quadrature position with the primary current. INDUCTION MACHINES 311 The polyphase induction motor of the three-phase or quarter- phase type is the one most commonly used, while single-phase motors have found a more limited application only, and especially for smaller powers. Thus in the following more particularly the polyphase induc- tion machine shall be treated, and the single-phase type discussed only in so far as it differs from the typical polyphase machine. 2. CALCULATION 136. In the polyphase induction motor, Let Y = g jb = primary exciting admittance, or admit- tance of the primary circuit with open secondary circuit; that is, ge = magnetic power current, be = wattless magnetizing current, where e = counter-generated e.m.f. of the motor; Z Q = r + jx Q = primary self -inductive impedance, and Zi = 7*1 + jxi = secondary self-inductive impedance, reduced to the primary by the ratio of turns. 1 All these quantities refer to one primary circuit and one corre- sponding secondary circuit. Thus in a three-phase induction motor the total power, etc., is three times that of one circuit, in the quarter-phase motor with three-phase armature 1J^ of the three secondary circuits are to be considered as corresponding to each of the two primary circuits, etc. Let e = primary counter-generated e.m.f., or e.m.f. generated in the primary circuit by the flux interlinked with primary and secondary (mutual induction); s = slip, with the primary fre- quency as unit; that is, s = denoting synchronous rotation, s = l standstill of the motor. We then have 1 s = speed of the motor secondary as fraction of syn- chronous speed, sf = frequency of the secondary currents, where / = frequency impressed upon the primary; 1 The self -inductive reactance refers to that flux which surrounds one of the electric circuits only, without being interlinked with the other circuits. 312 ELEMENTS OF ELECTRICAL ENGINEERING hence, . se = e.m.f. generated in the secondary. The actual impedance of the secondary circuit at the frequency sf is Zi 8 = 7*1 +jsxi; hence, the secondary current is se se where the primary exciting current is /oo =eY = e[g jb], and the total primary current is /o = e I (ai -f g) j (a 2 + b] where The e.m.f. consumed in the primary circuit by the impedance Z Q is /oZo, the counter-generated e.m.f. is e, hence, the primary terminal voltage is EQ = e + I Q Z Q = e[l + (bi j& 2 ) (r + jx )] .= e (ci jc 2 ), where Ci = 1 + robi + Xobz and c 2 = r 6 2 Xobi. Eliminating complex quantities, we have EQ = e Vci 2 + c 2 2 , hence, the counter-generated e.m.f. of motor, e = == , where EQ = impressed e.m.f., absolute value. Substituting this value in the equations of /i, /oo, /o, etc , gives the complex expressions of currents and e.m.fs., and elimi- nating the imaginary quantities we have the primary current, /o = e V&i 2 + 6 2 2 , etc. INDUCTION MACHINES 313 The torque of the polyphase induction motor (or any other motor or generator) is proportional to the product of the mutual magnetic flux and the component of ampere-turns of the sec- ondary, which is in phase with the magnetic flux in time, but in quadrature therewith in direction or space. Since the generated e.m.f. is proportional to the mutual magnetic flux and the num- ber of turns, but in quadrature thereto in time, the torque of the induction motor is proportional also to the product of the gen- erated e.m.f. and the component of secondary current in quadra- ture therewith in time and in space. Since /i = e (a\ ja 2 ) is the secondary current corresponding to the generated e.m.f. e, the secondary current in the quadrature position thereto in space, that is, corresponding to the e.m.f. je, is jli = e(a 2 and die is the component of this current in quadrature in time with the e.m.f. e. Thus the torque is proportional toe X die, or D = e z di n 2 + s*xi* ' (ex 2 + c 2 2 ) (n 2 + sV) This value D is in its dimension a power, and it is the power which the torque of the motor would develop at synchronous speed. 137. In induction motors, and in general motors which have a definite limiting speed, it is preferable to give the torque in the form of the power developed at the limiting speed, in this case synchronism, as "synchronous watts," since thereby it is made independent of the individual conditions of the motor, as its number of poles, frequency, etc., and made comparable with the power input, etc. It is obvious that when given in synchronous watts, the maximum possible value of torque which could be reached, if there were no losses in the motor, equals the power input. Thus, in an induction motor with 9000 watts power input, a torque of 7000 synchronous watts means that % of the maximum theoretically possible torque is realized, while the statement, "a torque of 30 pounds at 1-foot radius," would be meaningless without knowing the number of poles and the fre- quency. Thus, the denotation of the torque in synchronous 314 ELEMENTS OF ELECTRICAL ENGINEERING watts is the most general, and preferably used in induction motors. Since the theoretical maximum possible torque equals the power input, the ratio torque in synchronous watts output power input that is, actual torque maximum possible torque* is called the torque efficiency of the motor, analogous to the power efficiency or power output t power input that is, power output maximum possible power output * Analogously torque in synchronous watts volt-amperes input is called the apparent torque efficiency. The definitions of these quantities, which are of importance in judging induction motors, are thus: The "efficiency" or "power efficiency" is the ratio of the true mechanical output of the motor to the output which it would give at the same power input if there were no internal losses in the motor. The "apparent efficiency" or "apparent power efficiency" is the ratio of the mechanical output of the motor to the output which it would give at the same volt-ampere input if there were neither internal losses nor phase displacement in the motor. The "torque efficiency" is the ratio of the torque of. the motor to the torque which it would give at the same power input if there were no internal losses in the motor. The "apparent torque efficiency" is the ratio of the torque of the motor to the torque which it would give at the same volt- ampere input if there were neither internal losses nor phase dis- placement in the motor. The torque efficiencies are of special interest in starting where the power efficiencies are necessarily zero, but it nevertheless INDUCTION MACHINES 315 is of importance to find how much torque per watt or per volt- ampere input is given by the motor. Since D = e 2 ai is the power developed by the motor torque at synchronism, the power developed at the speed of (1 s) X synchronism, or the actual power output of the motor, is P = (1 - s) D = e 2 ai (1 - s) eViS (1 - s) The output P includes friction, windage, etc. ; thus, the net me- chanical output is P friction, etc. Since, however, friction, etc., depend upon the mechanical construction of the individual motor and its use, it cannot be included in a general formula. P is thus the mechanical output, and D the torque developed at the armature conductors. The primary current IQ = e (bi jb 2 ) has the quadrature components ebi and eb z . The primary impressed e.m.f. EQ = e (ci - jc 2 ) has the quadrature components eci and ec 2 . Since the components ebi and ec 2 , and eb 2 and eci, respectively, are in quadrature with each other, and thus represent no power, the power input of the primary circuit is eb l X eci + eb 2 X ec 2 , or P = e 2 (bid + 6 2 c 2 ). The volt-amperes or apparent input is obviously, P a + &2 2 ) (d 2 +c 2 2 ). 138. These equations can be greatly simplified by neglecting the exciting current of the motors, and approximate values of current, torque, power, etc., derived thereby, which are suffi- ciently accurate for preliminary investigations of the motor at speeds sufficiently below synchronism to make the total motor current large compared with the exciting current. 316 ELEMENTS OF ELECTRICAL ENGINEERING In this case the primary current equals the secondary current, that is, T se IQ = /i == -- = e (oi - ja 2 ), where and = etc. Oi+r) 2 + s 2 zi 2 139. Since the counter-generated e.m.f. e (and thus the im- pressed e.m.f. EQ) enters in the equation of current, magnetism, etc., as a simple factor, in the equations of torque, power input and output, and volt-ampere input as square, and cancels in the equation of efficiency, power-factor, etc., it follows that the current, magnetic flux, etc., of an induction motor are propor- tional to the impressed e.m.f., the torque, power output, power input, and volt-ampere input are proportional to the square of the impressed e.m.f., and the torque- and power efficiencies and the power-factor are independent of the impressed voltage. In reality, however, a slight decrease of efficiency and power- factor occurs at higher impressed voltages, due to the increase of resistance caused by the increasing temperature of the motor and due to the approach to magnetic saturation, and a slight decrease of efficiency occurs at lower voltages when including in the efficiency the loss of power by friction, since this is inde- pendent of the output and thus at lower voltage, that is, lesser output, a larger percentage of the output, so that the efficiencies and the power-factor can be considered as independent of the impressed voltage, and the torque and power proportional to the square thereof only approximately, but sufficiently close for many purposes. 3. LOAD AND SPEED CURVES 140. The calculation of the induction motor characteristics is most conveniently carried out in tabulated form by means of above-given equations as follows: Let Z Q = r + JXQ = 0.1 -f- 0.3 j = primary self-inductive im- pedance. Zi = TI + jxi = 0.1 + 0.3 j = secondary self-inductive impedance reduced to pri- mary. Y = g jb = 0.01 0.1 j = primary exciting admit- tance. EQ = 110 volts = primary impressed e.m.f. 318 ELEMENTS OF ELECTRICAL ENGINEERING It is then, per phase, 1 rO %, X n% [V* H| s ' + -0 *f it 1 ! \ h > *r : ? i* 0.0100 3 0.010.10 1.031 +0.007 1.031106.6 0.101010.8 0.01 o.oioolo.ioo 3.003 0.11 0.103 1.042 -0.0231.042 105.7 0.1507 15.9 0.02 0.0100 0.200 3.012 0.21 0.112 1.055 -0.052 1.056 104.3 0.238 24.8 0.05 0.0102 0.490 3.073 0.50 0.173 1.102 -0.133 1.110 99.20.522 51.8 0.1 0.0109 0.920 3.276 0.93 0.376 1.206 -0.241 1.230 89.5 1.003 89.7 0.15 0.2 0.0120 0.0136 1.25 1.47 3.563 3.883 1.26 1.48 0.663 0.983 1.325 1.443 -0.308 -0.354 1.360 1.485 80.91.424 74.21.777 115 132 0.3 0.5 1.0 0.01811.66 0.03251.54 O.lOOOjl.OO L.49 2.31 5.00 1.671.50 1.552.41 1.013.10 1.617 1.878 2.031 -0.351 -0.224 +0.007 1.654 1.891 2.031 66.6 58.2 54.1 2.245 2.865 3.261 149 167 176 s e* D = e 2 oi P = Pa = Eol p = bici + &2C2 Po 2 = eff. = P Po app. eff. = P Pa pow.fac. = Po Pa 11,360 1.19 0.011 0.125 10.5 0.01 11,170 1.117 1. 106 1.75 0.112 1.249 88.5 63.2 71.5 0.02 10,880 2.17C . 2. 133 2.73 0.216 2.350 91.0 78.3 86.2 0.05 9,840 4.82 4. 58 5.70 0.528 5.20 88.3 80.5 91.3 0.1 8,010 7.38 6. 64 9.87 1.030 8.25 80.7 67.3 83.5 0.15 6,540 8.20 6. 97 12.65 1.466 9.60 72.5 55.0 76.0 0.2 5,510 8.10 6. 48 14.52 1.782 9.80 66.0 44.6 67.5 0.3 4,440 7.36 5. 15 16.4 2.154 9.55 53.8 31.5 58.3 0.5 3,390 5.23 2. 61 18.4 2.370 8.04 32.3 14.2 43.8 1.0 2,930 2.93 19.4 2.072 6.08 31.3 Diagrammatically it is most instructive in judging about an induction motor to plot from the preceding calculation 1st. The load curves, that is, with the load or power output as abscissas, the values of speed (as a fraction of synchronism), of current input, power-factor, efficiency, apparent efficiency, and torque. 2d. The speed curves, that is, with the speed, as a fraction of synchronism, as abscissas, the values of torque, current input, power-factor, torque efficiency, and apparent torque efficiency. The load curves are most instructive for the range of speed near synchronism, that is, the normal operating conditions of the motor, while the speed curves characterize the behavior of the motor at any speed. INDUCTION MACHINES 319 In Fig. 176 are plotted the load curves, and in Fig 177 the speed curves of a typical polyphase induction motor of moderate size, having the folio wing constants: e Q = 110; Y = 0.01 0.1 j; Z, = 0.1 + 0.3 j, and Z = 0.1 + 0.3 j. As sample of a poor motor of high resistance and high admit- tance or exciting current are plotted in Fig. 178 the load curves of a motor having the following constants: e Q = 110; Y = 0.04 Z =Zf=0. 1+0.3 j .Y 0.01 -0.1 j 2000 3000 4000 5000 FIG. 176. Induction motor load cooo curves. - 0.4 j', Zi = 0.3 + 0.3 j, and Z Q = 0.3 + 0.3 j, showing the overturn of the power-factor curve frequently met in poor motors. 141. The shape of the characteristic motor curves depends entirely on the three complex constants, Y, Zi, and Z Q , but is essentially independent of the impressed voltage. Thus a change of the admittance Y has no effect on the char- acteristic curves, provided that the impedances Z\ and Z are 320 ELEMENTS OF ELECTRICAL ENGINEERING changed inversely proportional thereto, such a change merely representing the effect of a change of impressed voltage. A 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 FIG. 177. Induction motor speed curves. 1000 2000 3000 4000 FIG. 178. Load curves of poor induction motor. moderate change of one of the impedances has relatively little effect on the motor characteristics, provided that the other impedance changes so that the sum Zi + Z Q remains constant, INDUCTION MACHINES 321 and thus the motor can be characterized by its total internal impedance, that is, Z = Z l + Z ; or r + jx = (ri + r ) + j (xi + X Q ). Thus the characteristic behavior of the induction motor de- pends upon two complex imaginary constants, Y and Z, or four real constants, g, 6, r, x, the same terms which characterize the stationary alternating-current transformer on non-inductive load. Instead of conductance g, susceptance 6, resistance r, and react- ance x, as characteristic constants may be chosen: the absolute exciting admittance y = \/g 2 -f- & 2 ; the absolute self-inductive impedance z \/r 2 -}-x 2 ', the power-factor of admittance = g/y, and the power-factor of impedance a = r/z. 142. If the admittance y is reduced rz-fold and the impedance z increased n-fold, with the e.m.f. \SnE Q impressed upon the motor, the speed, torque, power input and output, volt-ampere input and excitation, power-factor, efficiencies, etc., of the motor, that is, all its characteristic features, remain the same, as seen from above given equations, and since a change of impressed e.m.f. does not change the characteristics, it follows that a change of admittance and of impedance does not change the character- istics of the motor provided the product 7 = yz remains the same. Thus the induction motor is characterized by three constants only: The product of exciting admittance and self-inductive impe- dance 7 = yz, which may be called the characteristic constant of the motor. The power-factor of exciting admittance /? = - y The power-factor of self-inductive impedance a = -- All these three quantities are absolute numbers. The physical meaning of the characteristic constant or the prod- uct of the exciting admittance and impedance is the following: If IQQ = exciting current and 7i = starting current, we have, approximately, E z = j-i MO Jf 00 y = y Z = ~ 322 ELEMENTS OF ELECTRICAL ENGINEERING The characteristic constant of the induction motor 7 = yz is the ratio of exciting current to starting current or current at standstill. At given impressed e.m.f., the exciting current 7oo is inversely proportional to the mutual inductance of primary and secondary circuit. The starting current Iio is inversely proportional to the sum of the self-inductance of primary and secondary circuit. Thus the characteristic constant 7 = yz is approximately the ratio of total self-inductance to mutual inductance of the motor circuits; that is, the ratio of the flux interlinked with only one circuit, primary or secondary, to the flux interlinked with both circuits, primary and secondary, or the ratio of the waste or leakage flux to the useful flux. The importance of this quantity is evident. 4. EFFECT OF ARMATURE RESISTANCE AND STARTING 143. The secondary or armature resistance TI enters the equa- tion of secondary current thus: S6 I STi . S*Xi -3 \ ) and the further equations only indirectly in so far as TI is con- tained in ai and a 2 . Increasing the armature resistance n-fold, to nri, we get at an n-fold increased slip ns, use se 1 " n + jsx that is, the same value, and thus the same values for e, Jo, D, Po, P a , while the power is decreased from P = (1 s) D to P = (1 ns) D, and the efficiency and apparent efficiency are correspondingly reduced. The power-factor is not changed; hence, an increase of armature resistance ri produces a propor- tional increase of slip s, and thereby corresponding decrease of power output, efficiency and apparent efficiency, but does not change the torque, power imput, current, power-factor, and the torque efficiencies. Thus the insertion of resistance in the armature or secondary of the induction motor offers a means of reducing the speed corresponding to a given torque, and thereby the desired torque can be produced at any speed below that corresponding to short- INDUCTION MACHINES 323 circuited armature or secondary without changing the input or current. Hence, given the speed curve of a short-circuited motor, the speed curve with resistance inserted in the armature can be derived therefrom directly by increasing the slip in proportion to the increased resistance. 1.0 0.9 08 SLIP FRACTION SLIP FRACTION C F 8YNC 00 05 04 03 02 0.1 Y 0.01 - 0. X HO TORQU WATT SS& 2000 1.0 0.9 0.8 0.7 0.0 0.5 0.4 0.3 0.2 0.1 FIG. 179. Induction motor speed-torque and -current curves. This is done in Fig. 179, in which are shown the speed curves of the motor Figs. 176 and 177, between standstill and syn- chronism, for Short-circuited armature, n = 0.1 (same as Fig. 177). 0.15 ohm additional resistance per circuit inserted in armature, r\ = 0.25, that is, 2.5 times increased slip. 324 ELEMENTS OF ELECTRICAL ENGINEERING 0.5 ohm additional resistance inserted in the armature, r\ = 0.6, that is, 6 times increased slip. 1.5 ohm 'additional resistance inserted in the armature, r\ = 1.6, that is, 16 times increased slip. The corresponding current curves are shown on the same sheet. With short-circuited secondary the maximum torque of 8250 synchronous watts is reached at 16 per cent. slip. The starting torque is 2950 synchronous watts, and the starting current 176 amp. With armature resistance TI = 0.25, the same maximum torque is reached at 40 per cent, slip, the starting torque is in- creased to 6050 synchronous watts, and the starting current decreased to 160 amp. With the secondary resistance r\ = 0.6, the maximum torque of 8250 synchronous watts approximately takes place in start- ing, and the starting current is decreased to 124 amp. With armature resistance 7*1 = 1.6, the starting torque is below the maximum, 5620 synchronous watts, and the starting current is only 64 amp. In the two latter cases the lower or unstable branch of the torque curve has altogether disappeared, and the motor speed is stable over the whole range; the motor starts with the maxi- mum torque which it can reach, and with increasing speed, torque and current decrease; that is, the motor has the character- istic of the direct-current series motor, except that its maximum speed is limited by synchronism. 144. It follows herefrom that high secondary resistance, while very objectionable in running near synchronism, is advantageous in starting or running at very low speed, by reducing the current input and increasing the torque. In starting we have s = 1. Substituting this value in the equations of subsection 2 gives the starting torque, starting current, etc., of the polyphase in- duction motor. In Fig. 180 are shown for the motor in Figs. 176, 177 and 179 the values of starting torque, current, power-factor, torque efficiency, and apparent torque efficiency for various values of the secondary motor resistance, from r\ = 0.1, the internal re- sistance of the motor, or R = additional resistance to n = 5.1 INDUCTION MACHINES 325 or R = 5 ohms additional resistance. The best values of torque efficiency are found beyond the maximum torque point. The same Fig. 180 also shows the torque with resistance in- serted into the primary circuit. The insertion of reactance, either in the primary or in the secondary, is just as unsatisfactory as the insertion of resistance in the primary circuit. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 FIG. 180. Induction motor starting torque with resistance secondary. in the Capacity inserted in the secondary very greatly increases the torque within the narrow range of capacity corresponding to resonance with the internal reactance of the motor, and the torque which can be produced in this way is far in excess of the maximum torque of the motor when running or when starting with resistance in the secondary. 326 ELEMENTS OF ELECTRICAL ENGINEERING But even at its best value, the torque efficiency available with capacity in the secondary is below that available with resistance. For further discussion of the polyphase inductance motor, see "Theory and Calculation of Alternating-current Phenomena." in. Single -phase Induction Motor 1. INTRODUCTION 146. In the polyphase motor a number of secondary coils displaced in position from each other are acted upon by a num- ber of primary coils displaced in position and excited by e.m.fs. displaced in phase from each other by the same angle as the dis- placement of position of the coils. In the single-phase induction motor a system of secondary circuits is acted upon by one primary coil (or system of primary coils connected in series or in parallel) excited by a single alter- nating current. A number of secondary circuits displaced in position must be used so as to offer to the primary circuit a short-circuited sec- ondary in any position of the armature. If only one secondary coil is used, the motor is a synchronous induction motor and belongs to the class of reaction machines. A single-phase induction motor will not start from rest, but when started in either direction will accelerate with increasing torque and approach synchronism. When running at or very near synchronism, the magnetic field of the single-phase induction motor is practically identical with that of a polyphase motor, that is, can be represented by the theory of the rotating field. Thus, in a turn wound under angle r to the primary winding of the single-phase induction motor, at synchronism an e.m.f. is generated equal to that generated in a turn of the primary winding, but differing therefrom by angle 6 = T in time phase. In a polyphase motor the magnetic flux in any direction is due to the resultant m.m.f. of primary and of secondary currents, in the same way as in a transformer. The same is the case in the direction of the axis of the exciting coil of the single-phase induc- tion motor. In the direction at right angles to the axis of the exciting coil, however, the magnetic flux is due to the m.m.f. of INDUCTION MACHINES 327 the secondary currents alone, no primary e.m.f. acting in this direction. Consequently, in the polyphase motor running synchronously, that is, doing no work whatever, the secondary becomes current- less, and the primary current is the exciting current of the motor only. In the single-phase induction motor, even when running light, the secondary still carries the exciting current of the mag- netic flux in quadrature with the axis of the primary exciting coil. Since, this flux has essentially the same intensity as the flux in the direction of the axis of the primary exciting coil, the current in the armature of the single-phase induction motor run- ning light, and therefore also the primary current corresponding thereto, has the same m.m.f., that is, the same intensity, as the primary exciting current, and the total primary current of the single-phase induction motor running light is thus twice the exciting current, that is, it is the exciting current of the main magnetic flux plus the current producing in the secondary the exciting current of the cross magnetic flux. In reality it is slightly less, especially in small motors, due to the drop of voltage in the self-inductive impedance and the drop of quadrature mag- netic flux below the impressed primary magnetic flux caused thereby. In the secondary at synchronism this secondary exciting current is a current of twice the primary frequency; at any other speed it is of a frequency equal to speed (in cycles) plus synchronism. Thus, if in a quarter-phase motor running light one phase is open-circuited, the current in the other phase doubles. If in the three-phase motor two phases are open-circuited, the current in the third phase trebles, since the resultant m.m.f. of a three- phase machine is 1.5 times that of one phase. In consequence thereof, the total volt-ampere input of the motor remains the same and at the same magnetic density, or the same impressed e.m.f., all induction motors, single-phase as well as polyphase, consume approximately the same volt-ampere input, and the same power input for excitation, and give the same distribution of magnetic flux. 146. Since the maximum . output of a single-phase motor at the same impressed e.m.f. is considerably less than that of a poly- phase motor, it follows therefrom that the relative exciting cur- rent in the single-phase motor must be larger. The cause of this cross magnetization in the single-phase indue- 328 ELEMENTS OF ELECTRICAL ENGINEERING tion motor near synchronism is that the secondary armature currents lag 90 deg. behind the magnetism, and are carried by the synchronous rotation 90 deg. in space before reaching their maximum, thus giving the same magnetic effect as a quarter- phase e.m.f. impressed upon the primary system in quadrature position with the main coil. Hence they can be eliminated by impressing a magnetizing quadrature e.m.f. upon an auxiliary motor circuit, as is done in the monocyclic motor. Below synchronism, the secondary currents are carried less than 90 deg., and thus the cross magnetization due to them is correspondingly reduced, and becomes zero at standstill. The torque is proportional to the power component of the armature currents times the intensity of magnetic flux in quad- rature position thereto. In the single-phase induction motor, the armature power currents I'\ = ea\ can exist only coaxially with the primary coil, since this is the only position in which corresponding pri- mary currents can exist. The magnetic flux in quadrature posi- tion is proportional to the component of e carried in quadrature, or approximately to (1 s) e, and the torque is thus D = (1 - s) el' = (1 - s) e 2 a lf thus decreases much faster with decreasing speed, and becomes zero at standstill. The power is then P = (1 - sYel' = (1 - s) 2 6 2 a!. Since in the single-phase motor only one primary circuit but a multiplicity of secondary circuits exist, all secondary circuits are to be considered as corresponding to the same primary cir- cuit, and thus the joint impedance of all secondary circuits must be used as the secondary impedance, at least at or near syn- chronism. Thus, if the armature has a quarter-phase winding of impedance Zi per circuit, the resultant secondary impedance is r? sr; if it contains a three-phase winding of impedance Zi per a 17 circuit, the resultant secondary impedance is =- In consequence hereof the resultant secondary impedance of a single-phase motor is less in comparison with the primary im- pedance than in the polyphase motor. Since the drop of speed under load depends upon the secondary resistance, in the single- INDUCTION MACHINES 329 phase induction motor the drop in speed at load is generally less than in the polyphase motor; that is, the single-phase induction motor has a greater constancy of speed than the polyphase induction motor, but just as the polyphase induction motor, it can never reach complete synchronism, but slips below synchro- nism, approximately in proportion to the speed. The further calculation of the single-phase induction motor is identical with that of the polyphase induction motor, as given in the previous chapter. Often no special motors are used for single-phase circuits, but polyphase motors adapted thereto. An induction motor with only one primary winding could not be started by a phase- splitting device, and would necessarily be started by external means. A polyphase motor, as for instance a three-phase motor operating single-phase, by having two of its terminals connected to the single-phase mains, is just as satisfactory a single-phase motor as one built with only one primary winding. The only difference is that in the latter case a part of the circumference of the primary structure is left without winding, while in the polyphase motor this part contains windings also, which, how- ever, are not used, or are not effective when running as single- phase motor, but are necessary when starting by means of displaced e.m.fs. Thus, in a three-phase motor operating from single-phase mains, in starting, the third terminal is connected to a phase-displacing device, giving to the motor the cross mag- netization in quadrature to the axis of the primary coil, which at speed is produced by the rotation of the secondary currents, and which is necessary for producing the torque by its action upon the secondary power currents. Thus the investigation of the single-phase induction motor resolves itself into the investigation of the polyphase motor operating on single-phase circuits. 2. LOAD AND SPEED CURVES 147. Comparing thus a three-phase motor of exciting admit- tance per circuit Y = g jb and self-inductive impedances ZQ = r Q + jx Q and Zi = TI + jxi per circuit with the same motor operating as single-phase motor from one pair of termi- nals, the single-phase exciting admittance is Y' = 3 Y (so as to give, the same volt-amperes excitation 3 eF), the primary 330 ELEMENTS OF ELECTRICAL ENGINEERING self-inductive impedance is the same, Z Q = r + jxo', the sec- ondary self-inductive impedance single-phase, however, is only y Z'i = -5-, since all three secondary circuits correspond to the same primary circuit, and thus the total impedance single-phase 17 is Z' = ZQ + -TT, while that of the three-phase motor is J = ZlQ "T" Zl\. Assuming approximately Z = Zi, we have Thus, in absolute value, s' = % 2, and T' = 2 T ; that is, the characteristic constant of a motor running single- phase is twice what it is running three-phase, or polyphase in 1000 2000 3000 4000 5000 7000 3000 9000 FIG. 181. Three-phase induction motor on single-phase circuit, load curves. general; hence, the ratio of exciting current to current at stand- still, or of waste flux to useful flux, is doubled by changing from polyphase to single-phase. This explains the inferiority of the single-phase motor com- pared with the polyphase motor. As a rule, an average polyphase motor makes a poor single- phase motor, and a good single-phase motor must be an excellent polyphase motor. INDUCTION MACHINES 331 As instances are shown in Figs. 181 and 182 the load curves and speed curves of the three-phase motor of which the curves of one circuit are given in Figs. 176 and 177, having the following constants : eo = 110 Three-phase Y = 0.01 -O.lj, Z Q = 0.1 + 0.3 j, Zi = 0.1 + 0.3,7, Thus, 7 = 6.36. Single-phase Y = 0.03 - 0.3 j, Z Q = 0.1 + 0.3 J, Z l = 0.033 +. 0.1 j, Thus, 7 = 12.72. It is of interest to compare Fig. 181 with Fig. 176 and to note the lesser drop of speed (due to the relatively lower secondary SLIP, S"= FIG. 182. Three-phase induction motor on single-phase circuit, s curves. resistance) and lower power-factor and efficiencies, especially at light load. The maximum output is reduced from 3 X 7000 = 21,000 in the three-phase motor to 9100 watts in the single-phase motor. Since, however, the internal losses are less in the single-phase motor, it can be operated at from 25 to 30 per cent, higher mag- netic density than the same motor polyphase, and in this case its output is from two-thirds to three-quarters that of the poly- phase motor. 148. The preceding discussion of the single-phase induction motor is approximate, and correct only at or near synchronism, 332 ELEMENTS OF ELECTRICAL ENGINEERING where the magnetic field is practically a uniformly rotating field of constant intensity, that is, the quadrature flux produced by the armature magnetization equal to the main magnetic flux produced by the impressed e.m.f. If an accurate calculation of the motor at intermediate speed and at standstill is required, the changes of effective exciting admittance and of secondary impedance, due to the decrease of the quadrature flux, have to be considered. At synchronism the total exciting admittance gives the m.m.f. of main flux and auxiliary flux, while at standstill the quad- rature flux has disappeared or decreased to that given by the starting device, and thus the total exciting admittance has de- 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 FIG. 183. Three-phase induction motor on single-phase circuit, torque curves. creased to one-half of its synchronous value, or one-half plus the exciting admittance of the starting flux. The effective secondary impedance at synchronism is the joint impedance of all secondary circuits; at standstill, however, only the joint impedance of the projections of the secondary coils on the direction of the main flux, that is, twice as large as at syn- chronism. In other words, from standstill to synchronism the effective secondary impedance gradually decreases to one-half its standstill value at synchronism. For fuller discussion hereof the reader must be referred to my second paper on the Single-phase Induction Motor, Transactions A. I. E. E., 1900, page 37. The torque in Fig. 182 obviously slopes toward zero at stand- INDUCTION MACHINES 333 still. The effect of resistance inserted in the secondary of the single-phase motor is similar to that in the polyphase motor in so far as an increase of resistance lowers the speed at which the maximum torque takes place. While, however, in the poly- phase motor the maximum torque remains the same, and merely shifts toward lower speed with the increase of resistance, in the single-phase motor the maximum torque decreases proportionally to the speed at which the maximum torque point occurs, due to the factor (1 s) entering the equation of the torque, D = e 2 ^ (1 - s). Thus, in Fig. 183 are given the values of torque of the single- phase motor for the same conditions and the same motor of which the speed curves polyphase are given in Fig. 179. The maximum value of torque which can be reached at any speed lies on the tangent drawn from the origin onto the torque curve for 7*1 = 0.1 or short-circuited secondary. At low speeds the torque of the single-phase motor is greatly increased by the insertion of secondary resistance, just as in the polyphase motor. 3. STARTING DEVICES OF SINGLE-PHASE MOTORS 149. At standstill, the single-phase induction motor has no starting torque, since the line of polarization due to the second- ary currents coincides with the axis of magnetic flux impressed by the primary circuit. Only when revolving is torque pro- duced, due to the axis of secondary polarization being shifted by the rotation, against the axis of magnetism, until at or near synchronism it is in quadrature therewith, and the magnetic disposition thus the same as that of the polyphase induction motor. Leaving out of consideration starting by mechanical means and starting by converting the motor into a series or shunt motor, that is, by passing the alternating current by means of commutator and brushes through both elements of the motor, the following methods of starting single-phase motors are left: 1st. Shifting of the axis of armature or secondary polarization against the axis of generating magnetism. 2d. Shifting the axis of magnetism, that is, producing a mag- netic flux displaced in position from the flux producing the arma- ture currents. 334 ELEMENTS OF ELECTRICAL ENGINEERING The first method requires a secondary system which is unsym- metrical in regard to the primary, and thus, since the secondary is movable, requires means of changing the secondary circuit, such as commutator brushes short-circuiting secondary coils in the position of effective torque, and open-circuiting them in the position of opposing torque. Thus this method leads to the repulsion motor, which is a commutator motor also. With the commutatorless induction motor, or motor with permanently closed armature circuits, all starting devices con- sist in establishing an auxiliary magnetic flux in phase with the secondary currents in time, and in quadrature with the line of secondary polarization in space. They consist in producing a component of magnetic flux in quadrature in space with the primary magnetic flux producing the secondary currents, and in phase with the latter, that is, in time quadrature with the primary magnetic flux. Thus, if F p = polarization due to the secondary currents, a = auxiliary magnetic flux, 6 = phase displacement in time between 3> a and 3> p , and T = phase displacement in space between ^a and F p , the torque is D = F p $ a sin T cos 6. In general the starting torque, apparent torque efficiency, etc., of the single-phase induction motor with any of these de- vices are given in per cent, of the corresponding values of the same motor with polyphase magnetic flux, that is, with a mag- netic system consisting of two equal magnetic fluxes in quad- rature in time and space. 150. The infinite variety of arrangements proposed for start- ing single-phase induction motors can be grouped into three classes. 1. Phase-splitting Devices. The primary system is composed of two or more circuits displaced from each other in position, and combined with impedances of different inductance factors so as to produce a phase displacement between them. When using two motor circuits, they can either be connected in series between the single-phase* mains, and shunted with impedances of different inductance factors, as, for instance, a INDUCTION MACHINES 335 condensance and an inductance, or they can be connected in shunt between the single-phase mains but in series with impe- dances of different inductance factors. Obviously the impe- dances used for displacing the phase of the exciting coils can either be external or internal, as represented by high-resistance winding in one coil of the motor, etc. In this class belongs the use of the transformer as a phase- splitting device by inserting a transformer primary in series with one motor circuit in the main line and connecting the other motor circuit to the secondary of the transformer, or by feeding one of the motor circuits directly from the mains and the other from the secondary of a transformer connected across the mains with its primary. In either case it is, respectively, the internal impedance, or internal admittance, of the transformer which is combined with one of the motor circuits for displacing its phase, and thus this arrangement becomes most effective by using transformers of high internal impedance or admittance as con- stant power transformers or open magnetic circuit transformers. 2. Inductive Devices. The motor is excited by the combina- tion of two or more circuits which are in inductive relation to each other. This mutual induction between the motor circuits can take place either outside of the motor in a separate phase- splitting device or in the motor proper. In the first case the simplest form is the divided circuit whose branches are inductively related to each other by passing around the same magnetic circuit external to the motor. In the second case the simplest form is the combination of a primary exciting coil and a short-circuited secondary coil on the primary member of the motor, or a secondary coil closed by an impedance. In this class belong the shading coil and the accelerating coil. 3. Monocyclic Starting Devices. An essentially wattless e.m.f. of displaced phase is produced outside of the motor, and used to energize a cross magnetic circuit of the motor, either directly by a special teaser coil on the motor, or indirectly by combining this wattless e.m.f. with the main e.m.f. and thereby deriving a system of e.m.fs. of approximately three-phase or any other relation. In this case the primary system of the motor is supplied essentially by a polyphase system of e.m.fs. with a single-phase flow of energy, a system which I have called "monocyclic." 336 ELEMENTS OF ELECTRICAL ENGINEERING The wattless quadrature e.m.f. is generally produced by con- necting two impedances of different inductance factors in series between the single-phase mains, and joining the connection between the two impedances to the third terminal of a three- phase induction motor, which is connected with its other two terminals to the single-phase lines, as shown diagrammatically in Fig. 184, for a conductance a and an inductive susceptance -jo,. This starting device, when using an inductance and a conden- sance of proper size, can be made to give an apparent starting torque efficiency superior to that of the polyphase induction motor. Usually a resistance and an inductance are used, which, though not giving the same starting torque efficiency as available by the use of a condensance, have the advantage of greater simplicity and cheapness. After starting, the impedances are disconnected. For a complete discussion and theoretical investigation of the FIG. 184. Connections for starting single-phase motor. different starting devices, the reader must be referred to the paper on the single-phase induction motor, American Institute of Electrical Engineers' Transactions, February, 1898." 151. The use of the resistance-inductance, or monocyclic, starting device with three-phase wound induction motor will be discussed somewhat more explicitly as the only method not us- ing condensers which has found extensive commercial application. It gives relatively the best starting torque and torque efficiencies. In Fig. 184, M represents a three-phase induction motor of which two terminals, 1 and 2, are connected to single-phase mains and the terminal 3 to the common connection of a conduct- ance a (that is, a resistance - j and an equal susceptance ja (thus a reactance H ) connected in series across the mains. Let Y = g jb = total admittance of motor between termi- INDUCTION MACHINES 337 nals 1 and 2 while at rest. We then have H Y = total admit- tance from terminal 3 to terminals 1 and 2, regardless of whether the motor is delta- or F-wound. If e = e.m.f. in the single-phase mains and E = difference of potential across conductance a of the starting device, then we have the current in a as /i = Ea, and the e.m.f. across ja as e E' thus, the current in ja is li = - ja (e - E), and the current in the cross magnetizing motor circuit from 3 to 1, 2 is /o = /i - /2 = Ea + ja (e - E). The e.m.f. ^ of the cross magnetizing circuit is, as may be seen from the diagram of e.m.fs., which form a triangle with e, E and e E as sides, Eo = E - (e - E) = 2 E - e, and since ! = H YE Q , we have Ea + ja (e - E) = % Y (2E - e). This expression solved for E becomes which from the foregoing value of E Q gives 3eaQ'+l) . ~3a-3ja-8F' or, substituting Y-g- jb, expanding, and multiplying both numerator and denominator by (3 a -80) +j(3a - 86), gives EQ = ea and the imaginary component thereof, or e.m.f. in quadrature to e in time and in space, is 338 ELEMENTS OF ELECTRICAL ENGINEERING In the same motor on a three-phase circuit this quadrature e.m.f. is the altitude of the equilateral triangle with e as sides, thus = je 7r-, and since the starting torque of the motor is pro- portional to this quadrature e.m.f., the relative starting torque of the monocyclic starting device, or the ratio of starting torque of the motor with monocyclic starting device to that of the same motor on three-phase circuit, is /)/ = EG! _ : 2a 2a- %( 1, corresponding to backward rotation of the ma- chine, the power input remains positive, the torque also remains positive, that is, in the same direction as for s < 1 ; but since the speed (I s) becomes negative or in opposite direction, the power output is negative, that is, the torque in opposite direc- tion to the speed. In this case the machine consumes electrical energy in its primary and mechanical energy by a torque oppos- ing the rotation, thus acting as brake. The total power, electrical as well as mechanical, is con- sumed by internal losses of the motor. Since, however, with large slip in a low-resistance motor the torque and power are small, the braking power of the induction machine at backward INDUCTION MACHINES 341 rotation is, as a rule, not considerable, excepting when using high resistance in the armature circuit. Z Zj- 0.1+ 0.3 j Y - 0.01 - 0.1 J 110 VOLTS CONSTANT FREQUENCY -1000 -2000 -3000 -4000 -5000 -6000 -7000 -8000 -9000 FIG. 185. Induction generator load curves. TORQUE POWER 8000 _0 loo; -4000 -6000 113 1.2 1U 1009 0,8 0:7 o!e 5 0!4 Oi3 ol2 0. ACTI A ^ SLIP FACTION OF SYNCHROS SM CONSTANT FREQUENCY CONSTANT TERMINAL VOLTAGE OF 110 Z - Y - 0.01 - 0.4 05 060 160 140 100. FIG. 186. Induction machine speed curves. Substituting for s negative values, corresponding to a speed above synchronism, torque and power output and power input 342 ELEMENTS OF ELECTRICAL ENGINEERING become negative, and a load curve can be plotted for the induc- tion generator which is very similar, but the negative counter- part of the induction motor load curve. It is for the machine shown as motor in Fig. 176 given as Fig. 185, while Fig. 186 gives the complete speed curve of this machine from s = 1.5 to * = -1. The generator part of the curve, for s < 0, is of the same char- acter as the motor part, s > 0, but the maximum torque and maximum output of the machine as generator are greater than as motor. Thus an induction motor when speeded up above synchronism acts as a powerful brake by returning energy into the lines, and the maximum braking effort and also the maximum electric power returned by the machine will be greater than the maxi- mum motor torque or output. 2. CONSTANT-SPEED INDUCTION OR ASYNCHRONOUS GENERATOR 154. The curves in Fig. 185 are calculated at constant fre- quency /, and thus to vary the output of the machine as gen- erator the speed has to be increased. This condition may be realized in case of induction generators running in parallel with synchronous generators under conditions where it is desirable that the former should take as much load as its driving power permits; as, for instance, if the induction generator is driven by a water power while the synchronous generator is driven by a steam engine. In this case the control of speed would be effected on the synchronous generator, and the induction gen- erator be without speed-controlling devices, running up beyond synchronous speed as much as required to consume the power supplied to it. Conversely, however, if an induction machine is driven at constant speed and connected to a suitable circuit as load, the frequency given by the machine will not be synchronous with the speed, or constant at all loads, but decreases with increasing load from practically synchronism at no load, and thus for the induction generator at constant speed a load curve can be con- structed as shown in Fig. 187, giving the decrease of frequency with increasing load in the same manner as the speed of the induction motor at constant frequency decreases with the load. In the calculation of these induction generator curves for con- INDUCTION MACHINES 343 slant speed the change of frequency with the load has obviously to be considered, that is, in the equations the reactance x has to be replaced by the reactance XQ (1 s), otherwise the equa- tions remain the same. FIG. 187. Induction generator load curves. 3. POWER-FACTOR OF INDUCTION GENERATOR 155. The induction generator differs 'essentially from a syn- chronous alternator (that is, a machine in which an armature revolves relatively through a constant or continuous magnetic field) by having a power-factor requiring leading current ; that is, in the synchronous alternator the phase relation between current and terminal voltage depends entirely upon the external circuit, and according to the nature of the circuit connected to the synchronous alternator the current can lag or lead the ter- minal voltage or be in phase therewith. In the induction or asynchronous generator, however, the current must lead the ter- minal voltage by the angle corresponding to the load and voltage of the machine, or, in other words, the phase relation between current and voltage in the external circuit must be such as required by the induction generator at that particular load. Induction generators can operate only on circuits with lead- ing current or circuits of negative effective reactance. 344 ELEMENTS OF ELECTRICAL ENGINEERING In Fig. 188 are given for the constant-speed induction gen- erator in Fig. 230 as function of the impedance of the external circuit z = -? as abscissas (where e Q = terminal voltage, i Q = 2o current in external circuit), the leading power-factor p = cos 6 required in the load, the inductance factor q = sin 6, and the frequency. Hence, when connected to a circuit of impedance z this induc- tion generator can operate only if the power-factor of its circuit is p', and if this is the case the voltage is indefinite, that is, the circuit unstable, even neglecting the impossibility of securing exact equality of the power-factor of the external circuit with that of the induction generator. FIG. 188. Three-phase induction generator power factor and inductance factor of external circuit. Two possibilities thus exist with such an induction generator circuit. 1st. The power-factor of the external circuit is constant and independent of the voltage, as when the external circuit consists of resistances, inductances, and capacities. In this case if the power-factor of the external circuit is higher than that of the induction generator, that is, the leading current less, the induction generator fails to excite and generate. If the power-factor of the external circuit is lower than that of the induction generator, the latter excites and its voltage rises until by saturation of its magnetic circuit and the consequent increase of exciting admittance, that is, decrease of internal power-factor, its power-factor has fallen to equality with that of the external circuit. INDUCTION MACHINES 345 In this respect the induction generator acts like the direct- current shunt generator, and gives load characteristics very similar to those of the direct-current shunt generator as dis- cussed in B; that is, it becomes stable only at saturation, but .Z^O.I + O.Sjl ATFULL V0.01 O.lj I FREQUENCY FIG. 189. Induction generator and synchronous motor load curves. loses its excitation and thus drops its load as soon as the voltage falls below saturation. Since, however, the field of the induction generator is alter- nating, it is usually not feasible to run at saturation, due to ex- cessive hysteresis losses, except for very low frequencies. 346 ELEMENTS OF ELECTRICAL ENGINEERING 2d. The power-factor of the external circuit depends upon the voltage impressed upon it. This, for instance, is the case if the circuit consists of a syn- chronous motor or contains synchronous motors or synchronous converters. In the synchronous motor the current is in phase with the impressed e.m.f. if the impressed e.m.f. equals the counter e.m.f. of the motor plus the internal loss of voltage. It is leading if the impressed e.m.f. is less, and lagging if the impressed e.m.f. is more. Thus when connecting an induction generator with a synchronous motor, at constant field excitation of the latter the 01 02 0,3 04 05 06 017 0.8 019 IjO 11. 12 13 14 l f .5 FIG. 190. Induction generator and synchronous converter, phase control, no line impedance. voltage of the induction generator rises until it is as much below the counter e.m.f. of the synchronous motor as required to give the leading current corresponding to the power-factor of the generator. Thus a system consisting of a constant-speed induc- tion generator and a synchronous motor at constant field excita- tion is absolutely stable. At constant field excitation of the synchronous motor", at no load the synchronous motor runs practically at synchronism with the induction generator, with a terminal voltage slightly below the counter e.m.f. of the syn- chronous motor. With increase of load the frequency and thus the speed of the synchronous motor drops, due to the slip of frequency in the induction generator, and the voltage drops, INDUCTION MACHINES 347 due to the increase of leading current required and the decrease of counter e.m.f. caused by the decrease of frequency. By increasing the field excitation of the synchronous motor with increase of load, obviously the voltage of the generator can be maintained constant, or even increased with the load. When running from an induction generator, a synchronous motor gives a load curve very similar to the load curve of an induction motor running from a synchronous generator; that is, a magnetizing current at no load and a speed gradually decreas- ing with the increase of load up to a maximum output point, at which the speed curve bends sharply down, the current curve upward, and the motor drops out of step. The current, however, in the case of the synchronous motor operated from an induction generator is leading, while it is lag- ging in an induction motor operated from a synchronous genera- tor. In either case it demagnetizes the synchronous machine and magnetizes the induction machine, that is, the synchronous machine supplies magnetization to the induction machine. In Fig. 189 is shown the load curve of a synchronous motor operated from the induction generator in Fig. 187. In Fig. 190 is shown the load curve of an over-compounded synchronous converter operated from an induction generator, the over-compounding being such as to give approximately constant terminal voltage e. 156. Obviously when operating a self-exciting synchronous converter from an induction generator the system is unstable also; if both machines are below magnetic saturation, since in this case in both machines the generated e.m,f. is proportional to the field excitation and the field excitation proportional to the voltage; that is, with an unsaturated induction generator the synchronous converter operated therefrom must have its mag- netic field excited to a density above the bend of the saturation curve. Since the induction generator requires for its operation a circuit with leading current varying with the load in the manner de- termined by the internal constants of the motor, to make an induction or asynchronous generator suitable for operation on a general alternating-current circuit, it is necessary to have a syn- chronous machine as exciter in the circuit consuming leading current, that is, supplying the required lagging or magnetizing current to the induction generator; and in this case the voltage 348 ELEMENTS OF ELECTRICAL ENGINEERING of the system is controlled by the field excitation of the syn- chronous machine, that is, its counter e.m.f. Either a synchro- nous motor of suitable size running light can be used herefor as exciter of the induction generator, or the exciting current of the induction generator may be derived from synchronous motors or converters in the same system, or from synchronous alternating- current generators operated in parallel with the induction gen- erator, in which latter case, however, these currents can be said to come from the synchronous alternator as lagging currents. Electrostatic condensers, as an underground cable system, may also be used for excitation, but in this case besides the condensers a synchronous machine or other means is required to secure stability. The induction machine may thus be considered as consuming a lagging reactive magnetizing current at all speeds, and con- suming a power current below synchronism, as motor, supplying a power current (that is, consuming a negative power current) above synchronism, as generator. Therefore, induction generators are best suited for circuits which normally carry leading currents, as synchronous motor and synchronous converter circuits, but less suitable for circuits with lagging currents, since in the latter case an additional syn- chronous machine is required, giving all the lagging currents of the system plus the induction generator exciting current. Obviously, when running induction generators in parallel with a synchronous alternator no synchronizing is required, but the induction generator takes a load corresponding to the excess of its speed over synchronism, or conversely, if the driving power behind the induction generator is limited, no speed regulation is required, but the induction generator runs at a speed exceeding synchronism by the amount required to consume the driving power. The foregoing consideration obviously applies to the polyphase induction generator as well as to the single-phase induction generator, the latter, however, requiring a larger exciter in con- sequence of its lower power-factor. Therefore, even in a single- phase induction generator, preferably polyphase excitation is used, that is, the induction machine and its synchronous exciter wound as polyphase machines, but the load connected to one phase only of the induction machine. The curves shown in the preceding apply to the machine as polyphase generator. The effect of resistance in the secondary is essentially the INDUCTION MACHINES 349 same in the induction generator as in the induction motor. An increase of resistance increases the slip, that is, requires an in- crease of speed at the same torque, current, and output, and thus correspondingly lowers the efficiency. Induction generators have been proposed and used to some extent for high-speed prime movers, as steam turbines, since their squirrel-cage rotor appears mechanically better suited for very high speeds than the revolving field of the synchronous generator. The foremost use of induction generators will probably be for collecting small water powers in one large system, due to the far greater simplicity, reliability, and cheapness of a small induction generator station feeding into a big system compared with a small synchronous generator station. The induction generator station requires only the hydraulic turbine, the induction ma- chine, and the step-up transformer, but does not even require a turbine governor, and so needs practically no attention, as the control of voltage, speed, and frequency takes place by a syn- chronous generator or motor main station, which collects the power while the individual induction generator stations feed into the system as much power as the available water happens to supply. The synchronous induction motor, comprising a single-phase or polyphase primary and a single-phase secondary, tends to drop into synchronism and then operates essentially as reaction machine. A number of types of synchronous induction genera- tors have been devised, either with commutator for excitation or without commutator and with excitation by low-frequency synchronous or commutating machine, in the armature, or by high-frequency excitation. For particulars regarding these very interesting machines, see " Theory and Calculation of Alternat- ing-current Phenomena." V. Induction Booster 157. In the induction machine, at a given slip s, current and terminal voltage are proportional to each other and of constant phase relation, and their ratio is a constant. Thus when con- nected in an alternating-current circuit, whether in shunt or in series, and held at a speed giving a constant and definite slip s, either positive or negative, the induction machine acts like a constant impedance. 350 ELEMENTS OF ELECTRICAL ENGINEERING The apparent impedance and its components, the apparent resistance and apparent reactance represented by the induction machine, vary with the slip. At synchronism apparent impe- dance, resistance, and reactance are a maximum. They decrease with increasing positive slip. With increasing negative slip the apparent impedance and reactance decrease also, the apparent FIG. 191. Effective impedance of three-phase induction machine. resistance decreases to zero and then increases again in negative direction as shown in Fig. 191, which gives the apparent impe- dance, resistance, and reactance of the machine shown in Figs, 176 and 177, etc., with the speed as abscissas. The cause is that the power current is in opposition to the ter- minal voltage above synchronism, and thereby the induction INDUCTION MACHINES 351 machine behaves as an impedance of negative resistance, that is, adding a power e.m.f. into the circuit proportional to the current. As may be seen herefrom, the induction machine when inserted in series in an alternating-current circuit can be used as a booster, that is, as an apparatus to generate and insert in the circuit an e.m.f. proportional to the current, and the amount of the boosting effect can be varied by varying the speed, that is, the slip at which the induction machine is revolving. Above synchronism the induction machine boosts, that is, raises the voltage; below synchronism it lowers the voltage; in either case also adding an out-of-phas.e e.m.f. due to its reactance. The greater the slip, either positive or negative, the less is the apparent resistance, positive or negative, of the induction machine. The effect of resistance inserted in the secondary of the induc- tion booster is similar to that in the other applications of the induction machine; that is, it increases the slip required for a certain value of apparent resistance, thereby lowering the effi- ciency of the apparatus, but at the same time making it less de- pendent upon minor variations of speed ; that is, requires a lesser constancy of slip, and thus of speed and frequency, to give a steady boosting effect. VI. Phase Converter 158. It may be seen from the preceding that the induction machine can operate equally well as motor, below synchronism, and as generator, above synchronism. In the single-phase induction machine the motor or generator action occurs in one primary circuit only, but in the direction in quadrature to the primary circuit there is a mere magnetizing current either in the secondary, in the single-phase motor proper, or in an auxiliary field-circuit, in the monocyclic motor. The motor and generator action can occur, however, simul- taneously in the same machine, some of the primary circuits acting as motor, others as generator circuits. Thus, if one of the two circuits of a quarter-phase induction machine is con- nected to a single-phase system, in the second circuit an e.m.f. is generated in quadrature with and equal to the generated e.m.f. in the first circuit; and this e.m.f. can thus be utilized to produce currents which, with currents taken from the primary single- phase mains, give a quarter-phase system. Or, in a three-phase motor connected with two of its terminals to a single-phase sys- 352 ELEMENTS OF ELECTRICAL ENGINEERING tern, from the third terminal an e.m.f. can be derived which, with the single-phase system feeding the induction machine, com- bines to a three-phase system. The induction machine in this application represents a phase converter. The phase converter obviously combines the features of a single-phase induction motor with those of a double transformer, transformation occurring from the primary or motor circuit to the secondary or armature, and from the secondary to the ter- tiary or generator circuit. Thus, in a quarter-phase motor connected to single-phase mains with one of its circuits, if Y = g jb = primary polyphase exciting admittance, ZQ = TQ -f- JXQ = self -inductive impedance per primary or ter- tiary circuit, Zi = ri + jxi = resultant single-phase self-inductive impe- dance of secondary circuits. Let e = e.m.f. generated by the mutual flux and Z = r + jx = impedance of the external circuit supplied by the phase converter as generator of second phase. We then have /> I = 71? current of second phase produced by phase Zr T Zo converter, E IZ = . = - ;=- = terminal voltage at genera- & -\- LQ 1 L o ~~z tor circuit of phase converter. The current in the secondary of the phase converter is then /! = / + /'+ I", where ^ I = load current = ~ I' = eY = exciting current of quadrature magnetic flux, S I' = - ; : = current required to revolve the machi ri+jsxi and the primary current is ?'-&> !', where /' = eY = exciting current of main magnetic flux. INDUCTION MACHINES 353 From these currents the e.m.fs. are derived in a similar manner as in the induction motor or generator. Due to the internal losses in the phase converter, the e.m.fs. of the two circuits, the motor and generator circuits, are prac- tically in quadrature with each other and equal only at no load, but shift out of phase and become more unequal with increase of load, the unbalancing depending upon the constants of the phase converter. An interesting application of the phase converter is made in single-phase induction motor railroading. In this, the phase converter is connected in series to the induction motor which drives the car. This avoids the increase of unbalanc- ing of the phases with increase of load, and makes it possi- ble by properly connected series transformers to maintain perfect phase and voltage balance on the driving motor. Usually, a quarter-phase phase converter and quarter-phase induction motor is used, and the motor phase of the phase converter is connected in series to one of the phases of the motor into the single-phase supply circuit, while the genera- tor phase of the phase converter feeds the other phase of the driving motor. It is obvious that the induction machine is used as phase con- verter only to change single-phase to polyphase, since a change from one polyphase system to another polyphase system can be effected by stationary transformers. A change from single- phase to polyphase, however, requires a storage of energy, since the power arrives as single-phase pulsating, and leaves as steady polyphase flow, and the momentum of the revolving phase con- verter secondary stores and returns the energy. With increasing load on the generator circuit of the phase converter its slip increases, but less than with the same load as mechanical output from the machine as induction motor. An application of the phase converter is made in single-phase motors by closing the tertiary or generator circuit by a condenser of suitable capacity, thereby generating the exciting current of the motor in the tertiary circuit. The primary circuit is thereby relieved of the exciting current of the motor, the efficiency essentially increased, and the power- factor of the single-phase motor with condenser in tertiary cir- cuit becomes practically unity over the whole range of load. At the same time, since the condenser current is derived by double 354 ELEMENTS OF ELECTRICAL ENGINEERING transformation in the multitooth structure of the induction machine, which has a practically uniform magnetic field, irre- spective of the shape of the primary impressed e.m.f. wave, the application of the condenser becomes feasible irrespective of the wave shape of the generator. Usually the tertiary circuit in this case is arranged on an angle of 60 deg. with the primary circuit, and in starting a powerful torque is thereby developed, with a torque efficiency superior to any other single-phase motor starting device, and when com- bined with inductive reactance in a second tertiary circuit, the apparent starting torque efficiency can be made even to exceed that of the polyphase induction motor (see page 336). For further discussion hereof, see A. I. E. E. Transactions, 1900, page 37. VH. Frequency Converter or General Alternating-current Transformer 159. The e.m.fs. generated in the secondary of the induction machine are of the frequency of slip, that is, synchronism minus speed, thus of lower frequency than the impressed e.m.f. in the range from standstill to double synchronism; of higher frequency outside of this range. ' Thus, by opening the secondary circuits of the induction machine and connecting them to an external or consumer's cir- cuit, the induction machine can be used to transform from one frequency to another, as frequency converter. It lowers the frequency with the secondary running at a speed between standstill and double synchronism, and raises the fre- quency with the secondary either driven backward or above double synchronism. Obviously, the frequency converter can at the same time change the e.m.f. by using a suitable number of primary and secondary turns, and can change the phases of the system by having a secondary wound for a different number of phases from the primary, as, for instance, convert from three phase 6000 volts 25 cycles to quarter phase 2500 volts 62.5 cycles. Thus, a frequency converter can be called a "general alter- nating-current transformer." For its theoretical discussion and calculation, see " Theory and Calculation of Alternating-current Phenomena." The action and the equations of the general alternating-current INDUCTION MACHINES 355 transformer are essentially those of the stationary alternating- current transformer, except that the ratio of secondary to primary generated e.m.f . is not the ratio of turns but the ratio of. the product of turns and frequency, while the ratio of secondary current and primary load current (that is, total primary current minus primary exciting current) is the inverse ratio of turns. The ratio of the products of generated e.m.f. and current, that is, the ratio of electric power generated in the secondary to electric power consumed in the primary (less excitation), is thus not unity but is the ratio of secondary to primary frequency. Hence, when lowering the frequency with the secondary re- volving at a speed between standstill and synchronism, the secondary output is less than- the primary input, and the differ- ence is transformed into mechanical work; that is, the machine acts at the same time as induction motor, and when used in this manner is usually connected to a synchronous or induction gen- erator feeding preferably into the secondary circuit (to avoid double transformation of its output) or to a synchronous con- verter, which transforms the mechanical power of the frequency converter into electrical power. When raising the frequency by backward rotation, the sec- ondary output is greater than the primary input (or rather the electric power generated in the secondary greater than the pri- mary power consumed by the generated e.m.f.), and the differ- ence is to be supplied by mechanical power by driving the fre- quency changer backward by synchronous or induction motor, preferably connected to the primary circuit, or by any other motor device. Above synchronism the ratio of secondary output to primary input becomes negative; that is, the induction machine generates power in the primary as well as in the secondary, the primary power at the impressed frequency, the secondary power at the frequency of slip, and thus requires mechanical driving power. The secondary power and frequency are less than the primary below double synchronism, more above double synchronism, and are equal at double synchronism, so that at double syn- chronism the primary and secondary may be connected in multi- ple or in series and the machine is then a double synchronous alternator further discussed in the "Theory and Calculation of Electrical Apparatus." As far as its transformer action is concerned, the frequency 356 ELEMENTS OF ELECTRICAL ENGINEERING converter is an open magnetic circuit transformer, that is, a trans- former of relatively high magnetizing current. It combines therewith, however, the action of an induction motor or generator. Excluding the case of over-synchronous rotation, it is approxi- mately (that is, neglecting internal losses) electrical input -r- electrical output -f- mechanical output = primary frequency -f- secondary frequency -f- speed or primary minus secondary fre- quency; that is, the mechanical output is negative when increas- ing the frequency by backward rotation. Such frequency converters are to a certain extent in com- mercial use, and have the advantage over the motor-generator plant of requiring an amount of apparatus equal only to the out- put, while the motor-generator set requires machinery equal to twice the output. An application of the frequency converter when lowering the frequency is made in concatenation or tandem control of induc- tion machines, as described in the next section. In this case the first motor, or all the motors except the last of the series are in reality frequency converters. VIII. Concatenation of Induction Motors 160. In the secondary of the induction motor an e.m.f. is generated of the frequency of slip. Thus connecting the sec- ondary circuit of the induction motor to the primary of a second induction motor, the latter is fed by a frequency equal to the slip of the first motor, and reaches its synchronism at the frequency of slip of the first motor, the first motor then acting as frequency converter for the second motor. If, then, two equal induction motors are rigidly connected together and thus caused to revolve at the same speed, the speed of the second motor, which is the slip s of the first motor at no load, equals the speed of the first motor: s = 1 s, and thus s = 0.5. That is, a pair of induction motors connected this way in tandem or in concatenation, that is, " chain connection/' as commonly called, or in cascade, as called abroad, tends to ap- proach s = 0.5, or half synchronism, at no load, slipping below this speed under load; that is, concatenation of two motors re- duces their synchronous speed to one-half, and thus offers as means to operate induction motors at one-half speed. In general, if a number of induction machines are connected INDUCTION MACHINES 357 in tandem, that is, the secondary of each motor feeding the primary of the next motor, the secondary of this last motor being short-circuited, the sum of the speeds of all motors tends toward synchronism, and with all motors connected together so as to revolve at the same speed the system operates at - synchronous speed, when n = number of motors. If the two induction motors on the same shaft have a different number of poles, they synchronize at some other speed below synchronism, or if con- nected differentially, they synchronize at some speed above synchronism. Assuming the ratio of turns of primary and secondary as 1 : 1, with two equal induction motors in concatenation at standstill, the frequency and the e.m.f. 'impressed upon the second motor, neglecting the drop of e.m.f. in the internal impedance of the first motor, equal those of the first motor. With increasing speed, the frequency and the e.m.f. impressed upon the second motor decrease proportionally to each other, and thus the magnetic flux and the magnetic density in the second motor, and its ex- citing current, remain constant and equal to those of the first motor, neglecting internal losses; that is, when connected in con- catenation the magnetic density, current input, etc., and thus the torque developed by the second motor, are approximately equal to those of the first motor, being less because of the internal losses in the first motor. Hence, the motors in concatenation share the work in approxi- mately equal portions, and the second motor utilizes the power which without the use of a second motor at less than one-half synchronous speed would have to be wasted in the secondary resistance; that is, theoretically concatenation doubles the torque and output for a given current, or power input into the motor system. In reality the gain is somewhat less, due to the second motor not being quite equal to a non-inductive resistance for the secondary of the first motor, and due to the drop of voltage in the internal impedance of the first motor, etc. At one-half synchronism, that is, the limiting speed of the con- catenated couple, the current input in the first motor equals its exciting current plus the transformed exciting current of the second motor, that is, equals twice the exciting current. 161. Henee, comparing the concatenated couple with a single motor, the primary exciting admittance is doubled. The total 358 ELEMENTS OF ELECTRICAL ENGINEERING impedance, primary plus secondary, is that of both motors, that is, doubled also, and the characteristic constant of the con- catenated couple is thus four times that of a single motor, but the speed reduced to one-half. FIG. 192. Comparison of concatenated motors with a single motor of double the number of poles. Comparing the concatenated couple with a single motor re- wound for twice the number of poles, that is, one-half speed also, such rewinding does not change the self-inductive impe- INDUCTION MACHINES 359 dance, but quadruples the exciting admittance, since one-half as many turns per pole have to produce the same flux in one-half the pole arc, that is, with twice the density. Thus the character- istic constant is increased fourfold also. It follows herefrom that the characteristic constant of the concatenated couple is that of one motor rewound for twice the number of poles. The slip under load, however, is less in the concatenated couple than in the motor with twice the number of poles, being due to only one-quarter the internal impedance, the secondary impedance of the second motor only, and thus the efficiency is slightly higher. 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -0.1-0.2-0.3-0.4-0.5-0.0-0.7 FIG. 193. Concatenation of induction motors, speed curves. Two motors coupled in concatenation are in the range from standstill to one-half synchronism approximately equivalent to one motor of twice the admittance, three times the primary impedance, and the same secondary impedance as each of the two motors, or more nearly 2.8 times the primary and 1.2 times the secondary impedance of one motor. Such motor is called the equivalent motor. 162. The calculation of the characteristic curve of the concate- nated motor system is similar to, but more complex than, that of the single motor. Starting from the generated e.m.f. e of the second motor, reduced to full frequency, we work up to the im- 360 ELEMENTS OF ELECTRICAL ENGINEERING pressed e.m.f. of the first motor e , by taking due consideration of the proper frequencies of the different circuits. Herefor the reader must be referred to " Theory and Calculation of Electrical Apparatus." The load curves of the pair of three-phase motors of the same constants as the motor in Figs. 176 and 177 are given in Fig. 192, the complete speed curve in Fig. 193. Fig. 192 shows the load curve of the total couple, of the two individual motors, and of the equivalent motor. As seen from the speed curve, the torque from standstill to one-half synchronism has the same shape as the torque curve of a single motor between standstill and synchronism. At one-half synchronism the torque reverses and becomes negative. It reverses again at about two-thirds synchronism, and is positive -0000 -4000 -2000 -2000 -4000 -000 , \ V z ,-0.1- -O.Sj Y - 3.01 - O.tj "~"x \ g a z A ^* 2. CO \\ .^ ^" V ""/ / 1.0 < .9 8 7 6 x, .5 ^ 4 3 2 1 0.( FIG. 194. Concatenation of induction motors speed curve with resistance in the secondary circuit. between about two-thirds synchronism and synchronism, zero at synchronism, and negative beyond synchronism. Thus, with a concatenated couple, two ranges of positive torque and power as induction motor exist, one from standstill to half synchronism, the other from about two-thirds synchro- nism to synchronism. In the ranges from one-half synchronism to about two-thirds synchronism, and beyond synchronism, the torque is negative, that is, the couple acts as generator. The insertion of resistance in the secondary of the second motor has in the range from standstill to half synchronism the same effect asin a single induction motor, that is, shifts the maxi- mum torque point toward lower speed without changing its value. Beyond half synchronism, however, resistance in the INDUCTION MACHINES 361 secondary lengthens the generator part of the curve, and makes the second motor part of the curve more or less disappear, as seen in Fig. 194, which gives the speed curves of the same motor as Fig. 193, with resistance in circuit in the secondary of the second motor. The main advantages of concatenation are obviously the abil- ity of operating at two different speeds, the increased torque and power efficiency below half speed, and the generator or braking action between half speed and synchronism, and such concatena- tion is therefore used to some extent in three-phase railway motor equipments, while for stationary motors usually a change of the number of poles by reconnecting the primary winding through a suitable switch is preferred where several speeds are desired, as it requires only one motor. INDEX Acceleration with starting device of single-phase induction motor, 338 Active component, 40 electromotive force, 39 Acyclic generator, 11 machine, 124 Adjustable speed alternating-current commutator motor, 221 Admittance, 98 exciting, of induction motor, 311 Air-blast transformer, 295 Air gap hi reactor, 303 Air reactors, 305 All day efficiency of transformer, 284 Alternating-current commutator motor, 218 generator electromotive force, 16 Angle of lag, 34 Apparent power efficiency, 314 torque efficiency, 314 Armature current of synchronous converter, 233 reaction coefficient, 208 of commutating machine, 193 of synchronous converter, 245 of synchronous machine, 130 resistance in induction motor, 322, 333 windings, 168 Asynchronous generator and motor, see "induction" Attenuation constant, 25 Auto-transformer, 124, 299 with direct-current converter, 263 with three-wire machines, 271 Average e.m.f., 13 Boosters, 122 induction, 349 Brush shift of commutating ma- chine, 181 Capacity, 54 Characteristic constant of induction motor, 321, 330 magnetic of commutating machine, 194 Charging current of condenser, 56 Choke coil, 124 Circuit, electric and magnetic, 2 Closed circuit winding, 171 Coefficient of armature reaction, 208 Commutating field, alternating-cur- rent motor, 220 machines, 121, 166 poles, 184 Commutation, 199 alternating-current motor, 219 Compensated alternating-current series motor, 221 Compensating winding of commu- tating machine, 190 Compensation for power-factor by alternating-current com- mutator motor, 221 Compensator, 124 also see "auto-transformer" synchronous, 123 Compound commutating machines, 166 generator, 213 motor, 216 Compounding action of commutat- ing poles, 188 curve of commutating machine, 196 of synchronous machine, 139, 144 of synchronous converter, 250 of transmission line, 90 Concatenation of induction motors, 256 363 364 INDEX Condensers, 54. 124 starting device of single-phase induction motor, 338 Condensive reactance, 55 Conductance, 100 Constant, characteristic of induc- tion motor, 321, 330 Converter, 122 direct-current, 262 synchronous, 223 three-Wire, 270 Core loss of transformer, 279 type transformer, 295 Counter e.m.f. of impedance, 35 of inductance, 32 of resistance, 33 Crank diagram, 41 Cross currents in synchronous ma- chines, 155 Cumulative compounding, 166, 217 Current, electric, 9 magnetic field, 2 ratio of converter, 230 Delta connection, 127 of transformer, 297 current, 127 voltage, 127 Demagnetization curve of separately excited or magneto ma- chine, 209 Demagnetizing armature reaction, 181 Diagram, crank or vector, 40 Dielectric field, 113 force, 113 hysteresis, 56 quantities, 116 Differential compounding, 166, 218 Direct-current converter, 262 generator e.m.f., 14 Distorting armature reaction, 181 Distortion of flux and saturation, 183 by armature reaction, 181 Division of load in parallel opera- tion of synchronous ma- chines, 154 Double-current generator, 123, 259 Double reentrant winding, 171 spiral winding, 171 Drum winding, 168 Dynamotors, 122 Eddy currents, 52 losses due to slots, 192 Effective reactance, 51 resistance, 48 values, 15 Effect of inductance, 28 Efficiency, 314 commutating machines, 198 reactors, 302 synchronous machines, 149 transformer, 280 all-day, 284 Electrical quantities, 115 symbols, 119 Electric circuit, 2 current, magnetic field, 2 Electrifying force, 117 Electrolytic apparatus, 122 Electromotive force, 113 consumed by impedance, 35 by inductance, 33 by resistance, 33 generated, 12 Electrostatic apparatus, 122 see "dielectric" Elimination of higher harmonics, 127 Energy, magnetic, 28 Equipotential surfaces, 115 Equivalent motor of concatenated induction motor, 358 sine waves, 106 Excitation of transformer, 279 Exciting admittance of induction motor, 311 current, 49 Field characteristic, commutating machine, 198 commutating, 185 of dielectric force, 113 of force, 112 of gravitational force, 110 intensity, 116 magnetic, 1, 5 INDEX 865 Field, magnetic, 1 of magnetic force, 113. Fluctuating cross current in syn- chronous machines, 155 Flux, magnetic, 1 of commutating machine, 178 Force, fields of, 112 lines, 115 mechanical in transformer, 294 Form factor, 126 of wave, 16 Foucault currents, 52 Fractional pitch winding, 175 Frequency of commutation, 200 converters, 124, 354 of synchronous converters, 257 Friction, molecular magnetic, 49 Full-pitch winding, 175 General alternating-current trans- former, 354 symbols, 120 wave, 107 Generated e.m.f., 12 of synchronous machine, 12& Gradient, 117 magnetic, 3 Gravimotive force, 113 Harmonics, higher, 127 third in transformer, 298 Heating in direct-current converter, 269 of synchronous converter, 233 of transformer, 294 Higher harmonics, 127 High-frequency cross current in syn- chronous machines, 159 High reactance transformer, 292 Hysteresis, 48 current, 50 dielectric, 56 lead, 50 resistance, 54 Impedance, 34, 98 counter e.m.f. of, 35 curves of transformer, 286 e.m.f. consumed by, 35 Impedance, of induction booster, 350 of induction motor, 311 of transmission line, 57 Inductance, 21 counter e.m.f., 32 e.m.f. consumed by, 33 Inductive devices, starting single- phase induction motor, 334 load, transmission line, 87 reactance, 32 Induction apparatus, stationary, 122 booster, 349 generator, 340, 343 machines, 122, 306 magnetic, 5 Instantaneous e.m.f., 13 Intensity of field, 113, 116 magnetic, 1, 5 Interpole, 186 Inverted converters, 123, 255 Lag angle, 34 Lap winding, 172 Lead, hysteretic, 50 Leading current as induction gen- erator load, 347 Leakage flux of transformer, 285 Level surfaces, 115 Line, also see "transmission line" of force, 115 reactance, 36 Load characteristic, series generator, 213 shunt generator, 210 transmission line, 85 curves of induction generator, 341 of induction motor, 317, 329 of synchronous machine, 141, 145 losses in synchronous machines. 150 saturation curve of commutat- ing machines, 196 Loaded transmission line, 57 Loss, commutating machines, 198 synchronous machines, 149 transformer, 280 366 INDEX Low reactance transformer, 292 Magnetic characteristic of synchron- ous machine, 147 circuit, 2 energy, 28 field, 1, 113 flux, 1 friction, molecular, 49 gradient, 3 hysteresis, 49 pole, 1 reaction, 10 symbols, 119 Magnetization curves, 8 of commutating machine, 194 Magnetizing current, 49 force, 3 Magneto generator, 209 machines, 166 Magnetomotive force, 2, 113 Maximum e.m.f., 13 power, transmission line, 86 Mechanical force, 113 in transformer, 294 Mohocyclic starting devices, single- phase induction motor, 334 Motive force, 113 Motor converter, 261 synchronous, 141 Multiple, also see "parallel" drum winding, 168 reentrant winding, 172 ring winding, 168 spiral winding, 171 Mutual inductance, 21 Neutral of commutating machine, 180 Nomenclature, 118 Nominal generated e.m.f., 128 Non-polar machine, 11, 124 Oil circulation in transformer, 295 cooled transformer, 294 Open-coil winding, 171 Output of direct-current converter, 269 of synchronous converter, 238 Over-compensation of armature re- action, 190 Over-compounding curve, commu- tating machine, 197 of transmission line, 93 Parallel connection of circuits, 101 operation of induction genera- tor, 348 of synchronous machines, 153 Parallelogram of sine wave, 44 Percentage saturation, 147, 195 Permeability, 5 Permittivity, 117 Phase, 45 characteristics of synchronous machines, 145 control by synchronous con- verter, 252 of transmission line, 90 converter, 123 induction, 350 splitting devices, starting single- phase induction motor, 334 Pitch deficiency and wave shape, 128 of winding, 175 Polarization cell, 124 ' Pole, magnetic, 1 strength, magnetic, 1 Polygon of sine waves, 44 Polyphase alternating-current com- mutator motor, 220 induction motor, 310 Potential regulator, 124 Power, 15, 39 component, 40 efficiency, 314 e.m.f., 39 factor of alternating-current commutator motor, 219 compensation by alternating- current commutator motor, 221 of induction generator, 343 Pulsation of armature reaction, 133 of flux by slots, 191 Quantity, electric, 118 Racing of inverted converter, 257 INDEX 367 Railway induction-phase converter, 353 Rating of direct-current converter, 269 of synchronous converter, 238 Ratio of transformer, 67, 277 Reactance, 99, 302 condensive, 55 effective, 51 inductive, 32 synchronous, 129, 136 of transformer, 285 of transmission line, 36 Reaction, armature, of commutating machine, 193 of synchronous converter, 245 of synchronous machine, 130 magnetic, 10 Reactive coil, 124, 302 component, 40 currents in synchronous con- verter, 250 e.m.f., 39 Reactor, 124, 302 Real generated e.m.f., 128 Rectangular coordinates, 77 Rectifying apparatus, 122 Reentrant winding, 171 Regulation of auto-transformer, 302 curve, commutating machine, 198 synchronous generator, 140 transformer, 285 transmission line, 57 Regulator, voltage, 124 Reluctance, 21 Repulsion motor, 221 in transformer, 294 Resistance, 9, 99 characteristic, series generator, 213 shunt generator, 211 commutation, 200 counter e.m.f., 33 effective, 48 e.m.f. consumed by, 33 leads, alternating-current motor, 220 in commutation, 206 Resistivity, 10 Ring connection, 127 winding, 168 Rotating magnetic field of induc- tion motor, 309 Saturation curve of commutating machine, 194 of synchronous machine, 147 effect of flux in commutating machine, 182 in synchronous machine, 148 factor of commutating machine. 195 of synchronous machine, 147 percentage, 147, 195 Self-inductance, 21 of commutation, 201 of synchronous machine, 133 of transformer, 285 Self-inductive impedance of induc- tion motor, 311 Self-induction, 10 Separately excited commutating machine, 166 generator, 209 Series alternating-current motor, 221 commutating machines, 166 connection of circuit, 101 drum "winding, 168 generator, 211 lap winding, 174 motor, 216 Shell-type transformer, 295 Shh\ of brushes of commutating machine, 181 Short circuit of auto-transformer, 302 currents of alternator, 160 of transformer, 293 loss of transformer, 286 Shunt-corn mutating machine, 166 generator, 210 motor, 215 Silicon steel, hysteresis, 52 Sine waves, equivalent, 106 Single-phase alternating -current commutator motors, 220 converter, 229 induction motor, 309, 326 368 INDEX Single-phase, short circuit of alter- nator, 162 Six-phase converter, 230 Size of auto-transformer, 300 of reactor, 303 Slip of frequency, induction genera- tor, 343 of induction booster, 350 of induction motor, 309 Slots, effect on magnetic flux, 190 Speed characteristic of series motor, 216 shunt generator, 211 shunt motor, 215 curves of induction generator, 341 of induction motor, 317, 329 of inverted converter, 256 Spiral winding, double and multiple, 171 Split-pole converter, 252 Stability of synchronous converter, 258 Star connection, 127 Starting of current, 24 devices of single-phase induction motor, 333 of induction motor, 3^2 of synchronous converter, 253 of synchronous motor, 151 Stopping of current, 26 Susceptance, 100 Symbols, 119 Symbolic method, 77 Synchronous commutating machine, 123 condenser, 123 converter, 223 induction generator and motor, 349 machines, 122, 126 motor starting, 151 reactance, 129, 136 watts, 313 Terminal voltage, 128 Third harmonic in transformer, 298 Three-phase transformer, 297 Three-wire converter, 270 generator, 270 Time constant, 25 Torque curves of induction motor, 323, 332 efficiency, 314 of induction motor, 313 maximum, of induction motor, 324 Transformer, 66, 77, 122, 277 general alterating current, 354 neutral, with three-wire con- verter, 275 Transmission line, compounding, 90 impedance, 57 load characteristic, 85 over-compounding, 93 phase control, 90 reactance, 36 Two circuit single-phase converter, 229 Types of transformers, 295 Unbalancing of polyphase synchron- ous machines, 150 Unipolar machine, 11, 124 Unit current, 2 e.m.f., 9 magnet pole, 1 Variable ratio converter, 252 Variation of converter voltage ratio, 231 Vector diagram, 41 Ventilation of transformer, 294 Virtual generated e.m.f., 128 Voltage commutation, 201 ratio of synchronous converter, 226 Volt ampere characteristic of re- actor, 304 Water circulation in transformer, 295 Wattless component, 40 e.m.f., 39 Wave winding, 172 Weight of mass, 113 Y connection, 127 of transformers, 297 current, 127 voltage, 127 Zero vector, 45 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DV OVERDUE. ENGINE "^' *tj^k '/ L\fl A W - ^ % 1949 ' OCT -22 1 ^V< V It i 10m-7,'44(1064s) YC 33427 UNIVERSITY OF CALIFORNIA LIBRARY